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Integrated Math 2 Section 8-4

The document discusses solving systems of equations by combining equations through addition or multiplication. It explains that combining equations can speed up the process of solving systems compared to graphing or substitution. An example problem demonstrates the steps: 1) choose a variable to eliminate, 2) make the coefficients opposite to combine equations, 3) solve the combined equation for one variable, 4) substitute back into the original equation to find the other variable. Checking the solution verifies the method works.

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Section 8-4 Solve Systems by Adding, Subtracting, and Multiplying (Linear Combinations)
Essential Questions How do you solve systems of equations by adding and subtracting? How do you solve systems of equations by adding and multiplying? Where you’ll see this: Landscaping, construction, sports
Why another method?
Why another method? Graphing: Problems with accuracy
Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable
Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable Combinations: Could speed up process
Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable Combinations: Could speed up process Look at the system to determine best method
Solve by Combinations
Solve by Combinations 1. Choose a variable to eliminate (your choice).
Solve by Combinations 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations.
Solve by Combinations 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable.
Solve by Combinations 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable.
Solve by Combinations 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable. 5. Check and rewrite the answer.
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18 
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20 10 10
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20 10 10 y=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 10 y = 20 10 10 y=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 10 10 y=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 y=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 y=2 −2 −2 x=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 4 + 14 = 18 y=2 −2 −2 x=2
Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 4 + 14 = 18 y=2 −2 −2 x=2 (2, 2)
Example 1 Solve by combining the equations 10x + 4 y = 14  b.  x + 4 y = 5 
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) (
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9 9 9
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9 9 9 x=1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 9 9 x=1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 x=1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 x=1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 x=1 y =1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 y =1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 1+ 4 = 5 y =1
Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 1+ 4 = 5 y =1 (1,1)
Example 2 Solve by combining the equations 7x + 2 y = 5    2x + 3 y = 16 
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16 
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) (
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17 x = −1
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 17x = −17 17 17 x = −1
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 17 17 x = −1
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 x = −1
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 y=6
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6
Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6 (−1, 6)
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3 
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) (
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3 x = 10
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 x = 10
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 x = 10
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 2 2 −43 y= 2
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 −43 y= 2
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 40 − 43 = −3 −43 y= 2
Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 40 − 43 = −3 −43 y= 2 (10, − ) 43 2
Homework
Homework p. 350 #1-27 odd “A verbal contract isn’t worth the paper it’s written on.” - Samuel Goldwyn

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Integrated Math 2 Section 8-4

  • 1.
    Section 8-4 Solve Systemsby Adding, Subtracting, and Multiplying (Linear Combinations)
  • 2.
    Essential Questions Howdo you solve systems of equations by adding and subtracting? How do you solve systems of equations by adding and multiplying? Where you’ll see this: Landscaping, construction, sports
  • 3.
  • 4.
    Why another method? Graphing: Problems with accuracy
  • 5.
    Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable
  • 6.
    Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable Combinations: Could speed up process
  • 7.
    Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable Combinations: Could speed up process Look at the system to determine best method
  • 8.
  • 9.
    Solve by Combinations 1.Choose a variable to eliminate (your choice).
  • 10.
    Solve by Combinations 1.Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations.
  • 11.
    Solve by Combinations 1.Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable.
  • 12.
    Solve by Combinations 1.Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable.
  • 13.
    Solve by Combinations 1.Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable. 5. Check and rewrite the answer.
  • 14.
    Example 1 Solve bycombining the equations −2x + 3 y = 2  a.   2x + 7 y = 18 
  • 15.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2
  • 16.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18
  • 17.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18
  • 18.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20
  • 19.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20 10 10
  • 20.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20 10 10 y=2
  • 21.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 10 y = 20 10 10 y=2
  • 22.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 10 10 y=2
  • 23.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 y=2
  • 24.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2
  • 25.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2
  • 26.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
  • 27.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
  • 28.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
  • 29.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
  • 30.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 y=2 −2 −2 x=2
  • 31.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 4 + 14 = 18 y=2 −2 −2 x=2
  • 32.
    Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 4 + 14 = 18 y=2 −2 −2 x=2 (2, 2)
  • 33.
    Example 1 Solve bycombining the equations 10x + 4 y = 14  b.  x + 4 y = 5 
  • 34.
    Example 1 Solve bycombining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) (
  • 35.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14
  • 36.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5
  • 37.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5
  • 38.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9
  • 39.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9 9 9
  • 40.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9 9 9 x=1
  • 41.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 9 9 x=1
  • 42.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 x=1
  • 43.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 x=1
  • 44.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1
  • 45.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
  • 46.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
  • 47.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
  • 48.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 x=1 y =1
  • 49.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 y =1
  • 50.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 1+ 4 = 5 y =1
  • 51.
    Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 1+ 4 = 5 y =1 (1,1)
  • 52.
    Example 2 Solve bycombining the equations 7x + 2 y = 5    2x + 3 y = 16 
  • 53.
    Example 2 Solve bycombining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16 
  • 54.
    Example 2 Solve bycombining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) (
  • 55.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15
  • 56.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32
  • 57.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32
  • 58.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17
  • 59.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17
  • 60.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17 x = −1
  • 61.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 17x = −17 17 17 x = −1
  • 62.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 17 17 x = −1
  • 63.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 x = −1
  • 64.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1
  • 65.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3
  • 66.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
  • 67.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
  • 68.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
  • 69.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
  • 70.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 y=6
  • 71.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6
  • 72.
    Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6 (−1, 6)
  • 73.
    Example 3 Solve bycombining the equations 5x + 2 y = 7   4x + 2 y = −3 
  • 74.
    Example 3 Solve bycombining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) (
  • 75.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7
  • 76.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3
  • 77.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3
  • 78.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3 x = 10
  • 79.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 x = 10
  • 80.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 x = 10
  • 81.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10
  • 82.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43
  • 83.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2
  • 84.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
  • 85.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
  • 86.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
  • 87.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 2 2 −43 y= 2
  • 88.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 −43 y= 2
  • 89.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 40 − 43 = −3 −43 y= 2
  • 90.
    Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 40 − 43 = −3 −43 y= 2 (10, − ) 43 2
  • 91.
  • 92.
    Homework p. 350 #1-27 odd “A verbal contract isn’t worth the paper it’s written on.” - Samuel Goldwyn