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**5.2: Solving Systems of Equations using Substitution**

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**Solving Systems of Equations using Substitution**

Steps: 1. Solve one equation for one variable (y= ; x= ; a=) 2. Substitute the expression from step one into the other equation, and SOLVE. 3. Substitute back into the equation we solved for in Step 1, and SOLVE 4. Check the solution in both equations of the system.

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Example #1: y = 4x 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 7x = -21 x = -3

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**y = 4x 3x + y = -21 Step 4: Substitute back into either original**

equation to find the value of the other variable. y = 4x = 4 (-3) y = -12 Solution to the system is (-3, -12).

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**y = 4x 3x + y = -21 3x + y = -21 y = 4x 3(-3) + (-12) = -21**

Step 5: Check the solution in both equations. Solution to the system is (-3,-12). 3x + y = -21 3(-3) + (-12) = -21 -9 + (-12) = -21 -21= -21 y = 4x -12 = 4(-3) -12 = -12

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**Example #2: x + y = 10 y = -x +10 5x - y = 2 5x -(-x +10) = 2**

Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5x - y = 2 5x -(-x +10) = 2

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**5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x = 2 x + y = 10**

Simplify!! 5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x = 2

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**Solution to the system is (2,8).**

x + y = 10 5x – y = 2 Step 4: Substitute back into the equation we solved for in step 1 y = -x + 10 = -(2) + 10 y = 8 Solution to the system is (2,8).

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**5x – y = 2 x + y =10 5(2) - (8) = 2 2 + 8 =10 10 – 8 = 2 10 =10 2 = 2**

Step 5: Check the solution in both equations. Solution to the system is (2, 8). 5x – y = 2 5(2) - (8) = 2 10 – 8 = 2 2 = 2 x + y =10 2 + 8 =10 10 =10

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**Solving a system of equations by substitution**

Pick the easier equation. The goal is to get y= ; x= ; a= ; etc. Step 1: Solve an equation for one variable. Step 2: Substitute Put the equation solved in Step 1 into the other equation. Substitute the value of the variable into the equation. Step 3: Plug back in to find the other variable. Step 4: Check your solution. Substitute your ordered pair into BOTH equations.

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**1) Solve the system using substitution**

x + y = 5 y = 3 + x Step 1: Solve an equation for one variable. The second equation is already solved for y! Step 2: Substitute x + y = 5 x + (3 + x) = 5 2x + 3 = 5 2x = 2 x = 1

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**1) Solve the system using substitution**

x + y = 5 y = 3 + x x + y = 5 (1) + y = 5 y = 4 Step 3: Plug back in to find the other variable. (1, 4) (1) + (4) = 5 (4) = 3 + (1) Step 4: Check your solution. The solution is (1, 4). What do you think the answer would be if you graphed the two equations?

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**Which answer checks correctly?**

3x – y = 4 x = 4y - 17 (2, 2) (5, 3) (3, 5) (3, -5)

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**2) Solve the system using substitution**

3y + x = 7 4x – 2y = 0 It is easiest to solve the first equation for x. 3y + x = 7 -3y y x = -3y + 7 Step 1: Solve an equation for one variable. Step 2: Substitute 4x – 2y = 0 4(-3y + 7) – 2y = 0

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**2) Solve the system using substitution**

3y + x = 7 4x – 2y = 0 -12y + 28 – 2y = 0 -14y + 28 = 0 -14y = -28 y = 2 4x – 2y = 0 4x – 2(2) = 0 4x – 4 = 0 4x = 4 x = 1 Step 3: Plug back in to find the other variable.

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**2) Solve the system using substitution**

3y + x = 7 4x – 2y = 0 Step 4: Check your solution. (1, 2) 3(2) + (1) = 7 4(1) – 2(2) = 0 When is solving systems by substitution easier to do than graphing? When only one of the equations has a variable already isolated (like in example #1).

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**2x + 4y = 4 3x + 2y = 22 -4y + 4 -2y + 2 -2x + 4 -2y+ 22**

If you solved the first equation for x, what would be substituted into the bottom equation. 2x + 4y = 4 3x + 2y = 22 -4y + 4 -2y + 2 -2x + 4 -2y+ 22

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**3) Solve the system using substitution**

x = 3 – y x + y = 7 Step 1: Solve an equation for one variable. The first equation is already solved for x! Step 2: Substitute x + y = 7 (3 – y) + y = 7 3 = 7 The variables were eliminated!! This is a special case. Does 3 = 7? FALSE! When the result is FALSE, the answer is NO SOLUTIONS.

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**3) Solve the system using substitution**

2x + y = 4 4x + 2y = 8 Step 1: Solve an equation for one variable. The first equation is easiest to solved for y! y = -2x + 4 4x + 2y = 8 4x + 2(-2x + 4) = 8 Step 2: Substitute 4x – 4x + 8 = 8 8 = 8 This is also a special case. Does 8 = 8? TRUE! When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS.

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