economists mig ht be described in terms of demand behavior, supply behavior,. So you see that for. Step 2: Choose 1 of those equations and the left over equation and do elimination. solution of a system of three linear equations. 3x3 system of equations solver This calculator solves system of three equations with three unknowns (3x3 system). To use elimination to solve a system of three equations with three variables, follow this procedure: Write all the equations in standard form cleared of decimals or fractions. 1 = - 10t + 2yt + 4zt. Otherwise it is independent. Engaging math & science practice! Improve your skills with free problems in 'Solving Systems of Linear Equations in Three Variables' and thousands of other practice lessons. Solve the system and answer the question. 5 Systems with Three Variables In lesson 3. 6: Solving Systems of Linear Equations in Three Variables (Terms). 3-6 LESSON TEKS 2A. Replace with 4 and with 3. Topics you will need to know to pass the quiz include. Solve for that variable. An example - where a, b, c and d are given constants, and both y and x are functions of t. Write down what each variable represents. System Of 3 Variable Equations - Displaying top 8 worksheets found for this concept. SOLVE the system of equations. The study of systems of polynomial equations in many variables requires a good understanding of what can be said about one polynomial equation in one variable. Nov 12, 2018. (If you have done everything correctly, this will produce one equation with one variable). There are TONS of fractions, so I hope you do not get drowned out with all the arithmetic!! Category. How would you solve a system of linear equations using C programming? Basically, you have 'x' many variables and 'x' many equations, where 'x' is a variable. Solution of a System In general, a solution of a system in three variables is an ordered triple ( x , y , z ) that makes ALL THREE equations true. Learn exactly what happened in this chapter, scene, or section of Systems of Three Equations and what it means. Systems of Linear Equations and Problem Solving. Outcome (learning objective) Students will accurately solve a system of equations algebraically using. If the system has an infinity number of solutions, it is dependent. 1 - Solving Systems of Equations. A "system" of equations is a set or collection of equations that you deal with all together at once. S OLVING S YSTEMS OF E QUATIONS IN T HREE V ARIABLES • Write out the 4 step process for solving for three unknowns, given a linear system of three equations. Solve a Simultaneous Set of Two Linear Equations This page will show you how to solve two equations with two unknowns. Possible Outcomes of Solving (Types of Systems) There are three possible outcomes when solving a. Solve Linear Systems with Three Variables by Elimination In this section, the elimination method is used to solve systems of three linear equations with three variables. Solving systems of linear equations online. 4x + y = 3. Type and execute this line before begining the project below. In other words, the basis variables form a parametric representation of the solution space. 3 4 Systems of Linear Equations in Three Variables Understand the geometry of systems of three equations in three variables. This is the three dimensional analogue of Section 14. First, look at the equations and see what possible combinations we might use. 4 Linear Programming 3. The outcomes for these systems of equations are: 1. where the numbers x, y, and z are solutions of •. Maria buys 5 pieces of clothing (some jeans and some sweaters), and pays $160. Label the remaining variables as free variables. The graphical method of solving a system of equations in three variables involves plotting the planes that are formed when graphing each equation in the system and then finding the intersection point of all three planes. Systems of Linear Equations. Next, take that number and plug it into the formula to solve for the other variable. When this is the case, we write and solve a system of equations in order to answer questions about the situation. S OLVING S YSTEMS OF E QUATIONS IN T HREE V ARIABLES • Write out the 4 step process for solving for three unknowns, given a linear system of three equations. Solving a Linear System by Substitution Step Procedure 1 Solve one equation for one of its variables. Eliminate the y. Tutorial 19: Solving a System of Linear Equations in Two Variables looked at three ways to solve linear equations in two variables. And here we have three equations with three unknowns. x + y + z + w = 0 x + y - z - w = 1 x - y + z + w = 3 x - y - z + w = -1 We can eliminate y, z and w simply by adding the second and third equations, term by term x + y - z - w = 1 x - y + z + w = 3 ----- 2x = 4 2x = 4 x = 2 Substitute 2 for x in the original equations. The outcomes for these systems of equations are: 1. (If you have done everything correctly, this will produce one equation with one variable). Systems of Linear Equations. 3 – Solving Systems of Equations by Elimination 2 Write your questions and thoughts here! MULTIPLYING AN EQUATION BY A CONSTANT TO ELIMINATE A VARIABLE e. A system of equations in three variables is inconsistent if no solution exists. Solve System of Linear Equations Using solve. notebook 3 September 24, 2013 elimination method Step 1: Choose 2 equations to use and do elimination to cancel out a variable. Step 2: Pick a different two equations and eliminate the same variable. Title: Solving Systems of Three Linear Equations in Three Variables 1 Solving Systems of Three Linear Equations in Three Variables. 3 Variable System of Equations. Maria buys 5 pieces of clothing (some jeans and some sweaters), and pays $160. If you solve for x, the result is x y 2. In this case everything is a first degree so we are dealing with a plane so we actually have here is three planes and we need to figure out how three planes can intersect okay?. Includes full solutions and score reporting. x + y + z = x + y + z = x + y. To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. Worksheet Assignment. notebook 3 September 24, 2013 elimination method Step 1: Choose 2 equations to use and do elimination to cancel out a variable. Now, we will work with more than variable and more than one equation. For example, the operation of the market for Ph. I cannot get Wolfram|alpha to solve a system (linear or otherwise) larger than 3 by 3. 3) Write the new system, complete with variables. 3 4 Systems of Linear Equations in Three Variables Understand the geometry of systems of three equations in three variables. Title: Solving Systems of Three Linear Equations in Three Variables 1 Solving Systems of Three Linear Equations in Three Variables. Example – Solve the system of equations by elimination 4x + 3y = –1 7x + 2y = 1. linear equations in two variables. System Of 3 Variable Equations - Displaying top 8 worksheets found for this concept. 3 3x 4y z 1 x y 4 z 3 x 3y 3z 9 5. Consider the prior example where x, the number of items produced and sold, was the independent variable in three functions, the cost function, the revenue function, and the profit function. Consider the same system of linear equations. (x, y, z) The solution must be a solution of all 3. More Examples Here are more examples of how to solve systems of equations in Algebra Calculator. The classification is straightforward -- an equation with n variables is called a linear equation in n variables. Apart from the stuff given in this section "Solving Linear Systems in Three Variables Worksheet", if you need any other stuff in math, please use our google custom search here. Nov 12, 2018. Each system has a number of equations and a number (not necessarily the same) of variables for which we would like to solve. In this lesson, we will learn about systems of 3 equations and 3 variables that may have a solution (x, y, z) that makes all three equations true. Reserved seat tickets for the football game cost $4. Systems of Linear Equations and Inequalities Recall that every linear equation in two variables can be identified with a line. This system will be equivalent to the given system, meaning that they share the same solution set. In Algebra II, sometimes we will be asked to solve systems of equations three variables. 1, Example 4(a) Solve graphically: y − x = 1, y + x = 3. 3 - 2: Solving Systems in Three Variables What is a system of 3 Equations with 3 Unknowns? A system of 3 equations with 3 unknowns is a list of three equations that have 3 variables in common. true true True statements result. If technology is present, using matrices is generally the quickest and most efficient. And, we can solve the first two equations to get x and y as functions of z alone. 5 Graphing Linear Equations in Three Variables 3. The general solution of the system of equations is: assign arbitrary values to the basis variables: read off the others from the equations for them. A triplet of numbers (x,y,z) is a solution of the system if EACH equation is satisfied by the triplet. Solve Systems of Three Equations in Three Variables Question Solve the system of equations in three variables. Use either. Videos, solutions, worksheets, games and activities to help Algebra students learn how to solve 3×3 systems of equations using the inverse of matrices. There are over 125 topics in all, from multi-step equations to trigonometric identities. Add the two equations together. In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. Pre-University Math. Solve system of 3 variable equations. In Lesson 7-1, you solved systems of linear equations graphically and algebraically. In the case below, each plane intersects the other two planes. A variable in mathematics is a symbol that is used in place of a value. Math 166, Fall 2011, c Benjamin Aurispa 4. 6 Solving Systems of Linear Equations in Three Variables Warm-Up. There is a simple tool for determining the number of solutions of a square system of equations: the determinant. The solution to the system is the set of points where all three planes intersect. In many of the examples below, I'll use the whole equation approach. They can have one point in common, just not all of them. Some of the worksheets for this concept are Systems of three equations elimination, Solving a system of two linear equations in two variables, , Solving a system of linear equations in three variables, Intermediate algebra skill solving 3 x 3 linear system by, Systems of three. Solve linear systems by graphing. Solve system of equations, no matter how complicated it is and find all the solutions. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. Solve the system of equations. We start with two pairs of equations and in each pair we eliminate the same variable. The point where the two lines intersect is called the “solution” of the. Solving Systems of Equations in 3 Variables Solving a system of equations is 3 variables isn’t much more complicated than a system of equations in two variables. Solving a system of linear equations can be done by hand, but it is a task that is time-consuming and error-prone. It's In the System. Do not use mixed numbers in your answer. Solve multiple equations by evaluating them in pairs % Solving Systems in Three Variables Using Elimination. an ordered triplet, 𝑥,𝑦,𝑧. Solving 3 variable systems of equations by substitution. Matrices are the perfect tool for solving systems of equations (the larger the better). Solve a Simultaneous Set of Two Linear Equations This page will show you how to solve two equations with two unknowns. Has anyone been able to do this? Yes, I have read all of the tutorials and I am familiar with the RREF command, but I’d just like to enter in the equations and see the solution. More Examples Here are more examples of how to solve systems of equations in Algebra Calculator. 3x3 system of equations solvers. Improve your math knowledge with free questions in "Solve a system of equations in three variables using elimination" and thousands of other math skills. economists mig ht be described in terms of demand behavior, supply behavior,. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. When solving systems of equations in three variables, it is generally recommended to use the elimination method to eliminate a variable. Here, we will solve systems with 2 variables, given in 2 linear equations. The strategy is to reduce this to two equations in two unknowns. These are the given equations. If the length of the list of variables doesn't match the number of linearly-independent equations to solve the result will be an empty list. When first introduced to systems of equations, you probably learned to solve a system of two-variable equations by graphing. Steps to solving a system of equations in 3 Variables. Usually, these questions have certain conditions. Arizona Department of Education. When we had two variables we reduced the system down to one with only one variable (by substitution or addition). And, we can solve the first two equations to get x and y as functions of z alone. The directions are from TAKS so do all three (variables, equations and solve) no matter what is asked in the problem. To create a function that returns a second derivative, one of the variables you give it has to be the first derivative. Methods of Solving a System of Equations in. Solving Systems of Equations by Substitution The substitution method for solving systems of linear equations in two variables can be extended to solve a system of linear equations in three variables. The solution to a system of three linear equations in three variables is an ordered triple. 1 Write corresponding augmented coe cient matrix 2 reduce to reduced row echelon form (rref), using three elementary row operations 3 from reduced matrix write the equivalent system of equations. Solve System of Linear Equations Using solve. Linear Equations in Three Variables To solve a systems of linear equations in three variables x , y , and z , we need to use the following steps: 1) Rewrite the systems of equations in three variables as systems of linear equations in two variables. If technology is present, using matrices is generally the quickest and most efficient. Solving Systems of Equations Algebraically Johnny Wolfe www. com is undoubtedly the right place to pay a visit to!. That solution will produce a true statement when substituted into every equation in the system. " Solving for 2 Unknowns-- Solve two equations in two unknowns. If the length of the list of variables doesn't match the number of linearly-independent equations to solve the result will be an empty list. 1 Solving Systems of Linear Equations in Three Variables Warm-Up No Solution Infinitely many solutions Here is a system of three linear equations in three variables: The ordered triple (2,-1,1) is a solution to this system since it is a solution to all three equations. Reserved seat tickets for the football game cost $4. Many word problems will give rise to systems of equations --- that is, a pair of equations like this: You can solve a system of equations in various ways. Example Solve the systems of equations (this example is also shown in our video lesson). Solve the system of linear equations and check any solution algebraically. 3 Graphing and Solving Systems of Linear Inequalities 3. Step 3: The results from steps one and two will each be an equation in two variables. There are TONS of fractions, so I hope you do not get drowned out with all the arithmetic!!. A system of equations involves one or more equations working together. Solving Systems of Three Equations w/ Substitution Date_____ Period____ Solve each system by substitution. Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 2; Linear System of Equations: Solving using Elimination by Addition; Solving a Dependent System of Linear Equations involving 3 Variables; Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 3. Systems of Equations Calculator is a calculator that solves systems of equations step-by-step. System of Linear Equations in three variables using Cramer’s Rule Cramer’s rule : In linear algebra, Cramer’s rule is an explicit formula for the solution of a system of linear equations with as many equations as unknown variables. dsolve can't solve this system. Many word problems will give rise to systems of equations --- that is, a pair of equations like this: You can solve a system of equations in various ways. Otherwise, a combination of the elimination and substitution methods works well. Can you help me solve this one? and explain everything please?. Find adequate exercises to solve a set of simultaneous equations with two variables using the graphing method and algebraic methods like the substitution method, elimination method, cross-multiplication method. The solution to a system of linear equations in three unknowns, if it exists, is. Systems of Linear Equations in Three Variables. Describe what you are doing at each step in your solution process. In other words, the basis variables form a parametric representation of the solution space. Gain immense practice with this batch of solving systems of equations worksheets, designed for high-school students. The problem is as follows: 2w - x + 5y + z = -3 3w + 2x + 2y - 6z = -32 w + 3x + 3y - z = -47 5w - 2x - 3y + 3z = 49 The direction for this problem say "solve the system of equations. Solving linear systems in three variables Algebra-expression. 1 2 3 0 4 2 1 9 3 0 Eq a b c Eq a b c Eq a b c + + = + + =− + + = Here you have a system of equations in 3 variables. The equations used are:. Solving Three Variable Equations by Substitution Method. (continued on the next page) Lesson 2-1 Solving Systems of Equations in Two Variables 69 Y ou can solve the second equation for either y or x. Class Notes. Solving systems of equations with 3 variables is very similar to how we solve sys-tems with two variables. 3-2 Bellwork - Solving Systems of Equations Algebraically. In the previous lesson, we discussed solving systems equation by substitution. systems of equations in three variables It is often desirable or even necessary to use more than one variable to model a situation in a field such as business, science, psychology, engineering, education, and sociology, to name a few. dsolve can't solve this system. Create a free account! When you create an account, we'll save your progress. Solving Large Systems of Equations To solve any system of linear equations , you will need to have at least the same number of independent equations as the number of variables. Can you help me solve this one? and explain everything please?. Eliminate the same variable from each pair using the Addition/Subtraction method. Systems of Equations Calculator is a calculator that solves systems of equations step-by-step. They can have one point in common, just not all of them. Tutorial 49: Solving a System of Linear Equations in Two Variables looked at three ways to solve linear equations in two variables. indd 24207c03-6_pr_TX. View Notes - 3. Algebra II Name: Worksheet #1 (§3 – 5) Solving Systems of Equations: Graphing Solve each system of equations by GRAPHING. Free linear equation worksheets that are useful both in higher mathematics and in everyday life. Note that at one point I say "term with no. Here is how it works. The method used in solving should be stated (the linear combination method was used to solve this system). like, finding integral solutions etc. This lesson teaches students how to solve a system of equations that contain three independent variables. Solving By Elimination: 3 equations in 3 variables Before we start on the next example, let's look at an improved way to do things. After having gone through the stuff given above, we hope that the students would have understood, "Solving Linear Systems in Three Variables Worksheet". Systems of linear equations solver Solving systems of linear equations with 3 and more variables is a difficult and time consuming task. Solving Systems of Equations with One Variable. Solving 3 x 3 systems of equations works a lot like the 2 x 2 method of elimination -- with a little substitution mixed in. 604 : Systems of Equations. indd 24 33/23/06 8:30:15 AM/23/06 8:30:15 AM. Session 1: Solving Systems of Equations usingGraphing (1 day) 1. 7 Name: _____ Solving Systems of Equations in Three Variables Date: _____ Homework Solve the following systems of equations. This Algebra Worksheet may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. But solving equations with three variables or more requires a new set of tricks, namely the techniques of elimination or substitution. In other words, we are looking for the ordered pairs ( x , y ) ( x , y ) that make both equations true. Do that by eliminating one of the unknowns from two pairs of equations: either from equations 1) and 2), or 1) and 3), or 2) and 3). 1 Write corresponding augmented coe cient matrix 2 reduce to reduced row echelon form (rref), using three elementary row operations 3 from reduced matrix write the equivalent system of equations. Algebra 2 Lesson 34 Solve Systems of Linear Equations in Three Variables. 8-3-Worksheet by Kuta Software LLC. Otherwise, a combination of the elimination and substitution methods works well. When solving these systems of equations, a 3D coordinate system is necessary since systems of equations with three variables are not linear. Let and Simplify. Linear equations are classified by the number of variables they involve. Attempt to solve the problem: Code (Text): def equations(p): y,z,t = p. Think back to linear equations. 4 Systems of Equations A linear equation in 2 variables is an equation of the form ax+ by = c. Solving One Variable Equations. Equations One-step equations Two-step equations Multi-step equations Absolute value equations Radical equations (easy, hard) Rational equations (easy, hard) Solving proportions Percent problems Distance-rate-time word problems Mixture word problems Work word problems Literal Equations. To solve real-life problems, such as finding the number of athletes who placed first, second, and third in a track meet in Ex. Look at the images below. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. This lesson teaches students how to solve a system of equations that contain three independent variables. Follow this method and we are less likely to make a mistake. And to do this, if we want to do it by elimination, if we want to be able to eliminate variables, it looks. To solve a system of N linear equations in N variables, select the number of equations, enter the coefficients of linear equations, then click "Solve". But it really doesn't matter that much in the overall effort required to solve the system. If a system of linear equations has at least one solution, it is consistent. 2 Substitute the expression from Step 1into the other equation and solve for the other variable. A solution of the system of equations is a point that lies on all three of those. Consider the nonlinear system. 3 Variable System of Equations. how to solve systems of of three variables equations using elimination? 0 votes-x - 5y - 5z=2 Solve each system in three variables using Gaussian elimination. LS 6: Solving Systems using Elimination LS 7: Solving Systems using Elimination (including reformatting equations) LS 8: Solving Systems Using Elimination (finding the least common multiple) LS 9: Systems with no solutions and infinitely many solutions (parallel and coincide) LS 10: Systems with Three equations and Three Variables. Tiger solves YOUR Linear Equations step by step using the Substitution Method, showing work{x+2y=1;3x-4y=13}. The whole problem with solving a system of equations is that you cannot solve an equation that has two unknowns in it. Add 4y to both sides of the equation, then divide the entire equation by -2. equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Three equations. The same situation occurs in three dimensions; the solution of 3 equations with 3 unknowns is the intersection of the 3 planes. Otherwise, a combination of the elimination and substitution methods works well. Put the equation in matrix form. Problem Solving in Two Variables Reference > Mathematics > Algebra > Systems of Equations Being able to solve systems of equations in two variables is great, but it's even better if you can use this skill in solving problems. How to solve a system of 3 equations and 3 variables using substitution to get two 2-variables equations, then elimination to solve for the 3 variables. Students will create a system of equations from a real world scenario and interpret the solution. Popular Tutorials in Solving Systems in Three Variables How Do You Solve a System of Equations in Three Variables Using Elimination? The elimination method can be extended from solving a system of equations with two variables to solving a system with three variables. The value of a variable depends on the other values in the expression or equation. Here are the 3 equation examples: x+2y+z=10. 2) used two of the equations to get rid of that variable by making the coefficients the same but opposite in sign. Questions 1. A linear equation in 3 variables is an equation of the form ax+ by + cz = d. System Of 3 Variable Equations - Displaying top 8 worksheets found for this concept. systems of equations in three variables It is often desirable or even necessary to use more than one variable to model a situation in a field such as business, science, psychology, engineering, education, and sociology, to name a few. Mathway Visit Mathway on the web. Apart from the stuff given in this section "Solving Linear Systems in Three Variables Worksheet", if you need any other stuff in math, please use our google custom search here. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. The study of systems of polynomial equations in many variables requires a good understanding of what can be said about one polynomial equation in one variable. How To Solve Systems Of 3 Variable Equations Using Elimination Step. like, finding integral solutions etc. Linear Combinations Method Substitution Method Solve the following system of equations: x - 2y = -10 y= 3x x - 2y = -10. 1) y = −4x + 16. A word problem is solved using a system of 3 equations with 3 variables. Solve the system of the two new equations using the Addition/Subtraction. Free Tutorials on how to solve equations, system of equations and inequalities using step by step approach with examples, detailed solutions and more exercises are presented. The method used in solving should be stated (the linear combination method was used to solve this system). Type and execute this line before begining the project below. where A, B, C, E, F, G, I, J, and K are the coefficients of the variables and D, H, and L are the values that the equations are equal to. You can put this solution on YOUR website! system of equation with four variables. High School Math Solutions - Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. Solving 3 variable systems of equations with no or infinite solutions. notebook 3 September 24, 2013 elimination method Step 1: Choose 2 equations to use and do elimination to cancel out a variable. From three variables graphing calculator online to number, we have got everything covered. x + 3y + z = -3 -x + y + 2z = -14 3x + 2y - z = 12 Solve the system and see that the sum of the fractions is 1 18. Practice Tests (Study Guides)!!! Systems of Equations Systems of Equations (Word Problems. Step 1- chose one of the equations and solve for a variable. Class Notes. And here we have three equations with three unknowns. Solving Systems of Equations in 3 Variables Solving a system of equations is 3 variables isn’t much more complicated than a system of equations in two variables. This is the three dimensional analogue of Section 14. I was trying to make them notice that they needed to make sure they were combining the same variables. These are called systems of equations. Here, a, b, and c are non - zero coefficients, d is a constant. Important Properties. Solving a system of two equations. the solution is the value of the three variables that make each equation true. No doubt you have solved systems of equations before. The matrix representing the system is 4 3 1. 2x + y + z = -2. Eliminate the y. Solve for the variables that have a leading one in their column. System solver can be used for solving systems of three linear equations in three variables or checking the solutions of 3 x 3 systems of linear equations solved by hand. In this video lesson, I will show you one easy method you can use to find each of your three solutions in a system of linear equations in three. As we saw in Section 8. Solve Systems of Three Equations in Three Variables Question Solve the system of equations in three variables. In this case I would try to eliminate the variable x first. 2x-3y-5z=27. Linear equations are classified by the number of variables they involve. The strategy is to reduce this to two equations in two unknowns. Systems Of Equations With 3 Variables. Variables can be letters, Greek symbols or combinations of many other symbols. equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. The directions are from TAKS so do all three (variables, equations and solve) no matter what is asked in the problem. { 3 2x y z 1 x 2y z 3 4x 3y 2z 5 3, 2, 2 1, 1, 2 Solve. Lesson 4 Linear Programming. Leave any comments or questions below. In other words, the basis variables form a parametric representation of the solution space. Hence, the above equation is a matrix equation. (x, y, z) The solution must be a solution of all 3. The goal is to reduce to 2 equations having 2 variables. SOLVE the system of equations. When solving systems of equations in three variables, it is generally recommended to use the elimination method to eliminate a variable. A system of two linear equations in two variables is of the form 𝑎𝑎+𝑥𝑥𝑏𝑏=𝑦𝑦𝑐𝑐 𝑑𝑑𝑥𝑥+ 𝑒𝑒=𝑦𝑦𝑓𝑓. A variable in mathematics is a symbol that is used in place of a value. To do this, you use row multiplications, row additions, or row switching, as shown in the following. Systems of 3 Equations in 3 Variables 1. 2x-3y-5z=27. These worksheets will fully prepare students for success in all topics from functions, graphing, and slopes, to inequalities and much more!. Find adequate exercises to solve a set of simultaneous equations with two variables using the graphing method and algebraic methods like the substitution method, elimination method, cross-multiplication method.