Solving Systems Using Elimination Section 6-3

Goals Goal To solve systems by adding or subtracting to eliminate a variable. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Elimination Method

Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two equations in the system together. Remember that an equation stays balanced if you add equal amounts to both sides. So, if 5x + 2y = 1, you can add 5x + 2y to one side of an equation and 1 to the other side and the balance is maintained. Elimination Method

Since –2y and 2y have opposite coefficients, the y- term is eliminated. The result is one equation that has only one variable: 6x = –18. When you use the elimination method to solve a system of linear equations, align all like terms in the equations. Then determine whether any like terms can be eliminated because they have opposite coefficients. Elimination Method

Solving Systems of Equations So far, we have solved systems using graphing and substitution. These notes show how to solve the system algebraically using ELIMINATION with addition and subtraction. Elimination is easiest when the equations are in standard form.

Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Add or subtract the equations. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Standard Form: Ax + By = C Look for variables that have the same coefficient. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations. Elimination Procedure

x + y = 5 3x – y = 7 Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. They already are! The y’s have the same coefficient. Step 3: Add or subtract the equations. Add to eliminate y. x + y = 5 (+) 3x – y = 7 4x = 12 x = 3 Example: Elimination by Addition

Step 4: Plug back in to find the other variable. x + y = 5 (3) + y = 5 y = 2 Step 5: Check your solution. (3, 2) (3) + (2) = 5 3(3) - (2) = 7 The solution is (3, 2). What do you think the answer would be if you solved using substitution? x + y = 5 3x – y = 7 Example: Continued

y + 3x = –2 2y – 3x = 14 Solve by elimination. Write the system so that like terms are aligned. 2y – 3x = 14 y + 3x = –2 Add the equations to eliminate the x-terms. 3y + 0 = 12 3y = 12 Simplify and solve for y. Divide both sides by 3. y = 4 Your Turn:

y + 3x = –2 Write one of the original equations x = –2 Substitute 4 for y. Subtract 4 from both sides. –4 3x = –6 Divide both sides by 3. 3x = –6 3 x = –2 Write the solution as an ordered pair. (–2, 4) Continued

3x – 4y = 10 x + 4y = –2 Solve by elimination. 3x – 4y = 10 Write the system so that like terms are aligned. Add the equations to eliminate the y-terms. 4x = 8 Simplify and solve for x. x + 4y = –2 4x + 0 = 8 Divide both sides by 4. 4x = 8 4 x = 2 Your Turn:

x + 4y = –2 Write one of the original equations y = –2 Substitute 2 for x. –2 4y = –4 4y –4 4 y = –1 (2, –1) Subtract 2 from both sides. Divide both sides by 4. Write the solution as an ordered pair. Continued

More Elimination When two equations each contain the same term, you can subtract one equation from the other to solve the system. To subtract an equation add the opposite of each term.

4x + y = 7 4x – 2y = -2 Step 1: Put the equations in Standard Form. They already are! Step 2: Determine which variable to eliminate. The x’s have the same coefficient. Step 3: Add or subtract the equations. Subtract to eliminate x. 4x + y = 7 (-) 4x – 2y = -2 3y = 9 y = 3 Remember to “keep-change- change” Example: Elimination by Subtracting

Step 4: Plug back in to find the other variable. 4x + y = 7 4x + (3) = 7 4x = 4 x = 1 Step 5: Check your solution. (1, 3) 4(1) + (3) = 7 4(1) - 2(3) = -2 4x + y = 7 4x – 2y = -2 Example: Continued

3x + 3y = 15 –2x + 3y = –5 Solve by elimination. 3x + 3y = 15 –(–2x + 3y = –5) 3x + 3y = x – 3y = +5 Add the opposite of each term in the second equation. Eliminate the y term. Simplify and solve for x. 5x + 0 = 20 5x = 20 x = 4 Your Turn:

Write one of the original equations. Substitute 4 for x. Subtract 12 from both sides. 3x + 3y = 15 3(4) + 3y = y = 15 –12 3y = 3 y = 1 Simplify and solve for y. Write the solution as an ordered pair. (4, 1) Continued

2x + y = –5 2x – 5y = 13 Solve by elimination. Add the opposite of each term in the second equation. –(2x – 5y = 13) 2x + y = –5 –2x + 5y = –13 Eliminate the x term. Simplify and solve for y y = –18 6y = –18 y = –3 Your Turn:

Write one of the original equations. 2x + y = –5 2x + (–3) = –5 Substitute – 3 for y. Add 3 to both sides. 2x – 3 = –5 +3 2x = –2 Simplify and solve for x. x = –1 Write the solution as an ordered pair. (–1, –3) Continued

Joke Time What do you call a man with no arms and no legs in a pool? Bob! What is T-Rex’s favorite number? Ate! How does the man in the moon cut his hair? Eclipse it!

Assignment 6-3 Part 1 Exercises Pg. 398: #6 – 20 even