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**The Substitution Method**

A method to solve a system of linear equations in 2 variables

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When you “Solve a system of equations” you are looking for a solution that will solve every equation in the system (group).

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**??? A linear equation Ax + By = C has an infinite number of solution? ???**

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How do we start with two equations , each having an infinite number solutions, and find the common solution (if any)???

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**This Method will create one combined equation with only one variable**

This Method will create one combined equation with only one variable. This is the kind of equation that you can solve!

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**Solve one equation for one of the variables**

Substitution Method - Step One Solve one equation for one of the variables Choose either equation and solve for either variable. (Choose the easiest one the solve. )

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**Step One - Solve one equation for one of the variables**

2x + y = 53 1 x + 5y = 139 2 Choose either equation and solve for either variable. (Choose the easiest one the solve. )

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**2x + y = 53 y = -2x +53 x + 5y = 139 x = - 5y +139 -2x -5y**

Step One - Solve one equation for one of the variables -2x In this problem you could have solved equation #1 for y or solved equation # 2 for x. 2x + y = 53 1 y = -2x +53 -5y x + 5y = 139 2 x = - 5y +139

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**Substitute this expression in the other equation and solve.**

Step One - Solve one equation for one of the variables Substitution Method - Step Two Substitute this expression in the other equation and solve.

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**2x + y = 53 y = -2x +53 x + 5y = 139 x + 5 (-2x +53) = 139**

Substitute this expression in the other equation and solve. If you solve for y in the first equation take this expression and substitute it in for y in the 2nd equation 2x + y = 53 1 y = -2x +53 x + 5y = 139 2 x + 5 (-2x +53) = 139 joined

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**This will create one combined equation with only one variable**

This will create one combined equation with only one variable. This is the kind of equation that you can solve!

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**x + 5 (-2x +53) = 139 x + -10x+265 = 139 -9x + 265 = 139 -9x = -126**

Substitute this expression in the other equation and solve. 2x + y = 53 y = -2x +53 Now solve for x 1 x + 5y = 139 2 x + 5 (-2x +53) = 139 joined x + -10x+265 = 139 -9x = 139 -9x = -126 x = -126/-9=14

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**Find the corresponding value of the other variable.**

After solving the combined equation …. Find the corresponding value of the other variable. (Substitute the value you found in step 2 to back into the equation)

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**y = -2x +53 y = -2(14)+53 (x , y ) = (14,25) y = -28+53=25 y = -2x +53**

Substitute the value you found for the first variable back into one of the original equations 2x + y = 53 1 x + 5y = 139 2 x + 5 (-2x +53) = 139 x + -10x+265 = 139 -9x = 139 -9x = -126 x = -126/-9 = 14 From the last step you found that x was 14. Take this value and plug it back into one of the original equations and find y. y = -2x +53 y = -2(14)+53 (x , y ) = (14,25) y = =25

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**3 ( ) + 2y = 113 3 (4y +5) + 2y = 113 12y +15+2y = 113 14y + 15 = 113**

Solve one of the equations for one of the variables x = 4y +5 x - 4y = 5 1 3x + 2y = 113 2 Substitute this expression into the other equation 3 ( ) + 2y = 113 3 (4y +5) + 2y = 113 12y +15+2y = 113 14y + 15 = 113 14y = 98 y = 98/14 = 7 and solve. Substitute the value you found for the first variable back into one of the original equations x = 4y +5 x = 4(7) +5 x = 28+5=33 (x , y ) = (33,7)

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**Step One - Solve one equation for one of the variables**

Step Two - Substitute this expression in the other equation and solve. Step Three - Find the other variable (Substitute value back into one of the equations)

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**I’m thinking of two numbers**

I’m thinking of two numbers. One number is one less then twice the other. The difference of the numbers is 18. Find the numbers.

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**I’m thinking of two numbers**

I’m thinking of two numbers. One number is one less then twice the other. The difference of the numbers is 18. Find the numbers. Let x and y represent the numbers. Write two equations to represent the relationships. y-x=18 y=2x-1

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**x-1 = 18 x = 19 (x , y ) = (19,37) y = 2x - 1 y = 2x-1 y - x = 18**

Solve one of the equations for one of the variables y = 2x - 1 y = 2x-1 1 y - x = 18 2 Substitute this expression into the other equation 2x x = 18 x-1 = 18 x = 19 and solve. Substitute the value you found for the first variable back into one of the original equations y = 2x - 1 y = 2(19) - 1 x = 38-1=37 (x , y ) = (19,37)

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**John had all dimes and quarters worth $5**

John had all dimes and quarters worth $5.45 If he had 35 coins in all, find out how many of each coin he had.

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**Let d = the # of dimes and q = the # of quarters Write two equations.**

John had all dimes and quarters worth $5.45 If he had 35 coins in all, find out how many of each coin he had. Let d = the # of dimes and q = the # of quarters Write two equations. d+q=35 10d+25q=545

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**10d + 25(35 - d) = 545 10d +875-25d = 545 -15d + 875 = 545 -15d = -330**

Solve one of the equations for one of the variables q = 35 - d d + q = 35 1 10d + 25q = 545 2 Substitute this expression into the other equation 10d + 25(35 - d) = 545 10d d = 545 -15d = 545 -15d = -330 d = -330/-15 = 22 and solve. q = 35 - d q = q = 13 Substitute the value you found for the first variable back into one of the original equations # of dimes = 22 # of quarters =13

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