A system of linear equations is a set of linear equations that have common variables. Common systems consist of two variables, x and y, and two linear equations. The solution to the system is the value of x and y that satisfy both equations. There are two ways to solve systems: substitution and elimination.
We will solve the following system in both ways.
1) x + y = 3
2) -3x + 5y = -1
Substitution
Step 1: Solve one equation in terms of one of the variables whichever is easier.
We rearrange equation 1 to form y = -x + 3.
Step 2: Substitute one equation into the other in terms of the variable found in step one.
-3x + 5(-x + 3) = -1
Step 3: Solve for the variable.
-3x – 5x + 15 = -1
-8x = -16
x = 2
Step 4: Substitute the value found back into one of the original equations and solve for the other variable.
x + y = 3
2 + y = 3
y = 1
So the solution to the system is x = 2 and y = 1.
Elimination
Step 1: Multiple one or both equations by a value so that the absolute value of one variable’s coefficients are the same.
Multiply (x + y = 3) by 3 to get 3x + 3x = 9
Now our equations are
1) 3x + 3y = 9
2) -3x + 5y = -1
Step 2: Add or subtract the sides of the equation to each other, whichever would eliminate the one variable with the same coefficients.
We are adding so that we can eliminate x.
3x + 3y + (-3x + 5y) = 9 + (-1)
Step 3: Simplify and solve for the remaining variable.
8y = 8
y = 1
Step 4: Substitute the found value back into the one of the original equations and solve for the other variable.
x + y = 3
x + 1 = 3
x = 2
So the solution to the system is x = 2 and y = 1.
This article was written for you by Jeremie, one of the tutors with Test Prep Academy.