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**SOLVING SYSTEMS USING SUBSTITUTION**

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Substitution is an algebraic model that can be used to find the exact solution of a system of equation. What does substitution mean? Give examples of where we have used substitution in Math before this unit.

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USING SUBSTITUTION What is the solution of the system? Use substitution. Check your answer by using your graphing calculator. y = 3x and x + y = - 32 Step 1: Because y = 3x, you can substitute 3x for y in x + y = - 32 x + y = Write the second equation. x + 3x = -32 Substitute 3x for y 4x = -32 Simplify x = - 8 Divide each side by 4 Step 2: Substitute – 8 for x in either equation and solve for y. y = 3x Write either equation. y = 3(- 8) = -24 Substitute – 8 for x and solve. The solution is (-8, - 24). Check by substituting (- 8, - 24) into each equation.

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**Let’s try one more together.**

What is the solution of the system? Use substitution. Check your answer by substituting your solution into each equation. Use your graphing calculator to check your answer. y = 2x + 7 and y = x - 1

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**With a partner, solve the following system using substitution**

With a partner, solve the following system using substitution. Show all work. Check your answer by graphing each equation on your graphing calculator. y = 3x x + y = 8

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**SOLVING FOR A VARIABLE AND USING SUBSTITUTION**

What is the solution of the system? Use substitution. 3y + 4x = 14 and -2x + y = -3 Step 1: Solve one of the equations for one of the variables. -2x + y = Write the second equation. -2x + 2x + y = -3 Add 2x to each side. y = 2x – 3 Simplify. Step 2: Substitute 2x – 3 for y in the other equation and solve for x. 3y + 4x = 14 Write the first equation. 3(2x – 3) + 4x = 14 Substitute 2x – 3 for y. Use parentheses. 6x– 9 + 4x = 14 Use the distributive property. 10x = 23 Add 9 to each side and simplify. x = Divide each side by 10.

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**Step 3: Substitute 2.3 for x in either equation and solve for y.**

-2x + y = Write either equation. -2(2.3) + y = Substitute 2.3 for x. y = Simplify. y = Add 4.6 to each side. The solution is (2.3, 1.6). Use your graphing calculator to check your answer.

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**Let’s try one more together.**

What is the solution of the system? Use substitution. Use your graphing calculator to check your answer. 6y + 5x = 8 x + 3y = - 7 Which variable did you solve for? Which equation did you use to solve for the variable?

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**With your partner, solve the system using substitution**

With your partner, solve the system using substitution. Check your answer by substituting and by using your graphing calculator. x + 3 = y 3x + 4y = 7

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**Work to solve these systems. Show all your work. Check your answer.**

y = - x – 2 3x + y = 12 y = 6x 2x + 3y = - 20 x = 2y + 7 x = y + 4

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Time to write…… When is the substitution method a better method than graphing for solving a system of linear equations? For each system, tell which equation you would first use to solve for a variable in the first step of the substitution method. Explain your choice. a. -2x + y = -1 and 4x + 2y = 12 b. x = 2y and 6x – y = 1 Tell whether each statement is true or false. Explain. a. When solving a system using substitution, if you obtain an identity (an equation with infinitely many solutions), then the system has no solution. b. You cannot use substitution to solve a system that does not have a variable with a coefficient of 1 or – 1.

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