Solving systems of linear equations is a skill not only taught in basic algebra but also tested on the ACT and SAT exams and even job application tests."Solving a system" is fancy language for finding the values of x and y that will make two equations true. "Substitution" means isolating one of the variables, x or y, in one of the equations, then "substituting" it, or plugging it in, to the other.
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Difficulty:
Moderate
Instructions
1
Understand that solving a "system" means you are working with two equations. An "equation" means that an equals sign separates two sides of a number sentence. Each equation will have two "variables," or unknown values. Usually, variables are identified with the letters x and y, though any letters can be used.For this example, we will work with the following two equations:2x + y = 812 + 3x = 10y + 1
2
Choose one of the equations to start with. The first step is isolating either x or y. Isolate means to set it on one side of the equals sign.Let's choose the first equation: 2x + y = 8.Since y has no coefficient, we will isolate it by moving 2x to the other side, shown below:2x + (-2x) + y = 8 - 2xy = 8 - 2xNow we have isolated y in the first equation.
3
Use the value for y in terms of x from our new Equation 1, and plug in the value of 8 - 2x for y in Equation 2:12 + 3x = 10(8 - 2x) + 1
4
Solve for the one variable, x, in Equation 2. Use the distributive property, as shown below:12 + 3x = 10(8 - 2x) + 112 + 3x = 80 - 20x + 1Combine like terms, as shown below:12 + 3x = 81 - 20xMove all of the terms including x on one side, as shown below:23x = 69Solve for x by dividing each side of the equation by 23:x = 3
5
Plug your known value of x = 3 into either equation and solve for y.Let's plug it back into Equation 1, 2x + y = 8:2(3) + y = 86 + y = 8y = 2Now we have solved for both x and y: x = 3 and y = 2. Double-check your work by plugging the values into both Equation 1 and Equation 2:Equation 1: 2(3) + 2 = 8 Equation 2: 12 + 3(3) = 10(2) + 1
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