**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:
- Point-Slope Form:
- Standard Form:

- 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.
- Write your answer as an ordered pair (x, y)

- 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)

- Graphing

- 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
- Use graphing
- Use substitution
- 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.

- If the variables have equal coefficients, use subtraction.

- 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 into either of the original equations to solve for y.
- 1st equation:
- Simplify the fraction:
- Divide the fraction:
- Subtract 3 from both sides:
- Divide by 7 on both sides:
- Answer:
- Check your answer using substitution

- 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!