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Warm Up Solve each equation for x. 1. y = x + 3 2. y = 3x – 4 Simplify each expression. x = y – 3 3. 2(x – 5) 2x – 10 4. 12 – 3(x + 1) 9 – 3x

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Warm Up Continued Evaluate each expression for the given value of x. 5. x + 8 for x = 6 6. 3(x – 7) for x =10 12 9

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Objective Solve linear equations in two variables by substitution.

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Sometimes it is difficult to identify the exact solution to a system by graphing. In this case, you can use a method called substitution. The goal when using substitution is to reduce the system to one equation that has only one variable. Then you can solve this equation by the methods taught in Chapter 2.

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**Solving Systems of Equations by Substitution**

Step 2 Step 3 Step 4 Step 5 Solve for one variable in at least one equation, if necessary. Step 1 Substitute the resulting expression into the other equation. Solve that equation to get the value of the first variable. Substitute that value into one of the original equations and solve. Write the values from steps 3 and 4 as an ordered pair, (x, y), and check.

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**Example 1A: Solving a System of Linear Equations by Substitution**

Solve the system by substitution. y = 3x y = x – 2 Step 1 y = 3x Both equations are solved for y. y = x – 2 Step 2 y = x – 2 3x = x – 2 Substitute 3x for y in the second equation. Step 3 –x –x 2x = –2 2x = –2 x = –1 Solve for x. Subtract x from both sides and then divide by 2.

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** Example 1A Continued Solve the system by substitution.**

Write one of the original equations. Step 4 y = 3x y = 3(–1) y = –3 Substitute –1 for x. Write the solution as an ordered pair. Step 5 (–1, –3) Check Substitute (–1, –3) into both equations in the system. y = 3x –3 3(–1) –3 –3 y = x – 2 –3 –1 – 2 –3 –3

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**Example 1B: Solving a System of Linear Equations by Substitution**

Solve the system by substitution. y = x + 1 4x + y = 6 The first equation is solved for y. Step 1 y = x + 1 Step 2 4x + y = 6 4x + (x + 1) = 6 Substitute x + 1 for y in the second equation. 5x + 1 = 6 Simplify. Solve for x. Step 3 –1 –1 5x = 5 x = 1 5x = 5 Subtract 1 from both sides. Divide both sides by 5.

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** Example1B Continued Solve the system by substitution.**

Write one of the original equations. Step 4 y = x + 1 y = 1 + 1 y = 2 Substitute 1 for x. Write the solution as an ordered pair. Step 5 (1, 2) Check Substitute (1, 2) into both equations in the system. y = x + 1 2 2 4x + y = 6 4(1) 6 6

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**Example 1C: Solving a System of Linear Equations by Substitution**

Solve the system by substitution. x + 2y = –1 x – y = 5 Step 1 x + 2y = –1 Solve the first equation for x by subtracting 2y from both sides. −2y −2y x = –2y – 1 Step 2 x – y = 5 (–2y – 1) – y = 5 Substitute –2y – 1 for x in the second equation. –3y – 1 = 5 Simplify.

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Example 1C Continued Step 3 –3y – 1 = 5 Solve for y. +1 +1 –3y = 6 Add 1 to both sides. –3y = 6 –3 –3 y = –2 Divide both sides by –3. Step 4 x – y = 5 Write one of the original equations. x – (–2) = 5 x + 2 = 5 Substitute –2 for y. –2 –2 x = 3 Subtract 2 from both sides. Write the solution as an ordered pair. Step 5 (3, –2)

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Check It Out! Example 1a Solve the system by substitution. y = x + 3 y = 2x + 5 Step 1 y = x + 3 y = 2x + 5 Both equations are solved for y. Step 2 2x + 5 = x + 3 y = x + 3 Substitute 2x + 5 for y in the first equation. –x – 5 –x – 5 x = –2 Step 3 2x + 5 = x + 3 Solve for x. Subtract x and 5 from both sides.

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**Check It Out! Example 1a Continued**

Solve the system by substitution. Write one of the original equations. Step 4 y = x + 3 y = –2 + 3 y = 1 Substitute –2 for x. Step 5 (–2, 1) Write the solution as an ordered pair.

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Check It Out! Example 1b Solve the system by substitution. x = 2y – 4 x + 8y = 16 Step 1 x = 2y – 4 The first equation is solved for x. (2y – 4) + 8y = 16 x + 8y = 16 Step 2 Substitute 2y – 4 for x in the second equation. Step 3 10y – 4 = 16 Simplify. Then solve for y. 10y = 20 Add 4 to both sides. 10y = Divide both sides by 10. y = 2

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**Check It Out! Example 1b Continued**

Solve the system by substitution. Step 4 x + 8y = 16 Write one of the original equations. x + 8(2) = 16 Substitute 2 for y. x + 16 = 16 Simplify. x = 0 – 16 –16 Subtract 16 from both sides. Write the solution as an ordered pair. Step 5 (0, 2)

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Check It Out! Example 1c Solve the system by substitution. 2x + y = –4 x + y = –7 Solve the second equation for x by subtracting y from each side. Step 1 x + y = –7 – y – y x = –y – 7 2(–y – 7) + y = –4 x = –y – 7 Step 2 Substitute –y – 7 for x in the first equation. 2(–y – 7) + y = –4 Distribute 2. –2y – 14 + y = –4

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**Check It Out! Example 1c Continued**

Solve the system by substitution. Step 3 –2y – 14 + y = –4 Combine like terms. –y – 14 = –4 –y = 10 Add 14 to each side. y = –10 Step 4 x + y = –7 Write one of the original equations. x + (–10) = –7 Substitute –10 for y. x – 10 = – 7

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**Check It Out! Example 1c Continued**

Solve the system by substitution. Step 5 x – 10 = –7 Add 10 to both sides. x = 3 Step 6 (3, –10) Write the solution as an ordered pair.

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Sometimes you substitute an expression for a variable that has a coefficient. When solving for the second variable in this situation, you can use the Distributive Property.

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When you solve one equation for a variable, you must substitute the value or expression into the other original equation, not the one that had just been solved. Caution

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**Example 2: Using the Distributive Property**

y + 6x = 11 Solve by substitution. 3x + 2y = –5 Solve the first equation for y by subtracting 6x from each side. Step 1 y + 6x = 11 – 6x – 6x y = –6x + 11 3x + 2(–6x + 11) = –5 3x + 2y = –5 Step 2 Substitute –6x + 11 for y in the second equation. 3x + 2(–6x + 11) = –5 Distribute 2 to the expression in parenthesis.

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Example 2 Continued y + 6x = 11 Solve by substitution. 3x + 2y = –5 Step 3 3x + 2(–6x) + 2(11) = –5 Simplify. Solve for x. 3x – 12x + 22 = –5 –9x + 22 = –5 –9x = –27 – 22 –22 Subtract 22 from both sides. –9x = –27 – –9 Divide both sides by –9. x = 3

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Example 2 Continued y + 6x = 11 Solve by substitution. 3x + 2y = –5 Write one of the original equations. Step 4 y + 6x = 11 y + 6(3) = 11 Substitute 3 for x. y + 18 = 11 Simplify. –18 –18 y = –7 Subtract 18 from each side. Step 5 (3, –7) Write the solution as an ordered pair.

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Check It Out! Example 2 –2x + y = 8 Solve by substitution. 3x + 2y = 9 Step 1 –2x + y = 8 Solve the first equation for y by adding 2x to each side. + 2x x y = 2x + 8 3x + 2(2x + 8) = 9 3x + 2y = 9 Step 2 Substitute 2x + 8 for y in the second equation. 3x + 2(2x + 8) = 9 Distribute 2 to the expression in parenthesis.

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**Check It Out! Example 2 Continued**

–2x + y = 8 Solve by substitution. 3x + 2y = 9 Step 3 3x + 2(2x) + 2(8) = 9 Simplify. Solve for x. 3x + 4x = 9 7x = 9 7x = –7 –16 –16 Subtract 16 from both sides. 7x = –7 Divide both sides by 7. x = –1

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**Check It Out! Example 2 Continued**

–2x + y = 8 Solve by substitution. 3x + 2y = 9 Write one of the original equations. Step 4 –2x + y = 8 –2(–1) + y = 8 Substitute –1 for x. y + 2 = 8 Simplify. –2 –2 y = 6 Subtract 2 from each side. Step 5 (–1, 6) Write the solution as an ordered pair.

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Lesson Quiz: Part I Solve each system by substitution. 1. 2. 3. y = 2x (–2, –4) x = 6y – 11 (1, 2) 3x – 2y = –1 –3x + y = –1 x – y = 4

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