Systems of Equations – Two Variable – Elimination

A system of equations is a set of two equations with two variables that we use to find an intersection point between two items. A system allows us to compare two items, analyze data and discover two unknowns.

There are several ways to solve systems of equations: graphing, substitution, elimination, Cramer’s Rule, and calculator.

Here we will focus on the elimination method.

Vocabulary

Coefficient – the number in front of the variable

How to Solve Systems of Equations using Elimination

Step 1: To use elimination, both equations need to have “x” and “y” on the same side of the equation. Make sure that “x” and “y” are on the same side before moving to step 2.

Step 2: Decide what variable you want to eliminate. It does not matter which variable you choose to eliminate. However, one might be easier than the other.

Step 3: Decide how you will eliminate. There are three ways you can do this.

A.  Add or Subtract – If the coefficients are the same for the variable you want to eliminate, subtract the equations. If the coefficients are opposites, add the equations.

B.  Multiply One Equation – If you notice that the coefficient in one equation is a multiple of the coefficient in the second equation, multiply the equation by the number that will make the coefficients the same, then subtract.

C.  Multiply Both Equations – If you do not notice the multiples, then you can multiply both equations. You would do this by multiplying the second equation by the coefficient of the variable you want to eliminate in the first equation, and then multiply the first equation by the coefficient of the second equation. This will make the coefficients the same, then subtract.

Step 4: Solve for the variable you have left.

Step 5: Substitute the value for the variable into one of the original equations and solve for the other variable.

Step 6: Write the solution as an ordered pair (x, y).

EXAMPLES:

1.  2x+y=63x-y=4 2. 4x+3y=124x-y=6

3.  5x+6y=11x-3y=2 4. 3x+4y=85x+7y=15

Special Cases

No Solution – If you add or subtract the two equations and you end up with “zero equals a number” there is no solution.

Example: 10x+14y=125x+7y=5

Infinite Number of Solutions – If you add or subtract the two equations and you end up with “zero equals zero” there is an infinite number of solutions.

Example: -18x+9y=-18-20x+10y=-20