# 5.3 – Two-Variable Systems: Elimination

Key Terms

• Elimination – The adding or subtracting of two equations to remove one of the variables in a system of equations.

Review

• Three Forms of Linear Equations
• Slope-Intercept Form: $y=mx+b$
• Point-Slope Form: $y-y_1=m(x-x_1)$
• Standard Form: $Ax+By=C$
• To solve a system of equations, you have already learned how to graph and use substitution.
• Graphing
• Use slope-intercept form to graph lines and see where they meet.
• Substitution
• Solve one equation for a variable, and substitute that variable’s expression into the other equation.
• Solve for both variables and write your answer as an ordered pair (x, y)

• Solving for a Variable
• When you “solve for x,” you find the value of x
• When you “solve for y,” you find the value of y

Notes

• Methods for Solving Systems of Linear Equations
1. Use graphing
2. Use substitution
3. Use elimination

• Elimination
• The elimination method is useful when you can eliminate one of the variable terms from an equation by adding or subtracting another equation.
• The elimination method works best if the equations are in standard form: Ax + By = C.

• Rules for Using Elimination
• If the variables have equal coefficients, use subtraction.
• Subtracting an equation is like multiplying every term in the subtracted equation by -1, and then adding the two equations.
• If the variables have opposite coefficients, use addition.

• Steps for Solving with Elimination
• 1. Find the variable with equal or opposite coefficients.
• 2. If equal, subtract the equations. If opposite, add the equations.
• 3. Solve the resulting one-variable equation.
• 4. Substitute the known value into the other equation and solve for the remaining variable.

• Example: solve using elimination
• 15x + 7y = 4
• 5x + 7y = 2

• So, subtracting the y-terms is going to help us eliminate the y’s.

• Substitute $x=\frac{1}{5}$ into either of the original equations to solve for y.
• 1st equation: $15(\frac{1}{5})+7y=4$
• Simplify the fraction: $\frac{15}{5}+7y=4$
• Divide the fraction: $3+7y=4$
• Subtract 3 from both sides: $3+7y=4$
• Divide by 7 on both sides: $\frac{7y}{7}=\frac{1}{7}$
• Answer: $y=\frac{1}{7}$

• Coefficients that Do Not Match
• If none of the variable terms have equal or opposite coefficients, multiply one or both equations by a number in order to create equal or opposite coefficients.
• See Examples 2, 3, and 4 (below)

More Examples

• Ex 1. Use elimination to solve
• 3x – 4y = -30
• 3x + 2y = 6
• Step 1: x is the variable with equal coefficients
• Step 2: Multiply the 2nd equation by -1 (using distribution)
• Step 3: Combine (add) the equations together
• Step 4: Since the x’s were eliminated, solve for y
• Step 5: Substitute y = 6 back into either of the original equations to solve for x
• Step 6: List the answer as an ordered pair (x, y)

• Ex 2. Use elimination to solve
• 2x – 3y = 11
• 5x + 4y = 16
• Step 1: You will have to distribute a number into EACH equation to eliminate the x or y terms.  For this example, we’ll choose x.
• Step 2: What is a multiple of 2 and 5?  10!  Let’s make 10x and -10x so we can eliminate the x-terms!
• Step 3: Multiply the top equation by positive 5 and multiply the bottom equation by -2.
• Step 4: Combine (add) the equations and solve for y.
• Step 5: Substitute y into either of the original equations to solve for x.
• Step 6: Write your answer as an ordered pair (x, y)

• Ex 3. What happens when  you eliminate both x and y?
• 6x + 9y = 15
• 2x + 3y = 5
• Step 1: Pick a variable to eliminate.  Let’s choose x for this example.
• Step 2: What is a multiple of 6 and 2?  6!  Let’s make 6x and -6x so we can eliminate the x-terms!
• Step 3: Distribute -3 into the 2nd equation.
• Step 4: Combine (add) both equations.
• Step 5: Uh, oh!  Both x and y were eliminated!
• Step 6: Since 0 = 0 (same answer on both sides), we have two lines on top of one another (same slope, same y-intercept)!  They have infinitely many solutions!  EVERY number you try for x or y will check out!

• Ex 4. What happens when you eliminate both x and y?
• 6x + 9y = 15
• 2x + 3y = 6
• Step 1: Pick a variable to eliminate. Let’s choose x for this example.
• Step 2: What is a multiple of 6 and 2? 6! Let’s make 6x and -6x so we can eliminate the x-terms!
• Step 3: Distribute -3 into the 2nd equation.
• Step 4: Combine (add) both equations.
• Step 5: Uh, oh! Both x and y were eliminated!
• Step 6: However, since 0 ≠ -3 (impossible), we have two parallel lines (same slope, different y-intercept)! They have NO solutions as they will never intersect!