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**5-3 Elimination Using Addition and Subtraction**

Algebra Glencoe McGraw-Hill Linda Stamper

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**When solving a linear system by substitution, you needed to isolate one of the variables.**

Sometimes it is not easy to isolate one of the variables. What is the result when you try to isolate one of the variables in the system below?

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When it is not easy to isolate one of the variables using the substitution method, it may be easier to solve the system by elimination. When using this method, you will be adding the two equations together in order to eliminate one of the variables.

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**Elimination Using Addition**

Add the equations if one of the variables has opposites. Then solve. Substitute the solved value into either of the original equations to find the value for the other variable. Write the solution as an ordered pair. Check by substitution in both original equations. (4,3)

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**Use elimination to solve the system of equations.**

Example 1 Example 2 Example 3 (1,2) (–2,3) (3,5)

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When the coefficients of terms are the same, you can eliminate the terms by subtracting the equations. When using elimination with subtraction to solve systems of equations, do not forget to distribute the negative sign over the entire equation that is subtracted. Since subtraction is the same as adding the inverse, you should change the signs of the terms and then add to eliminate the variable.

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**Elimination Using Subtraction**

Subtract the equations if one of the variables have the same coefficients. Then solve. - - - Substitute the solved value into either of the original equations to find the value for the other variable. Write the solution as an ordered pair. Check by substitution in both original equations. (4,-7)

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**It will not matter which equation you create the opposite!**

Elimination Using Subtraction Distribute the negative sign over the entire equation that is subtracted. - + + It will not matter which equation you create the opposite! (-1,2)

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**Use elimination to solve the system of equations.**

Example 4 Example 5 + - - - - - (2,6)

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**Write and Solve a System of Equations**

Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers. Assign Labels. Let x = first number Let y = second number Translate each sentence. Solve the system. Write a sentence to give the answer. The numbers are 5 and 8.

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**Example 6 The sum of two numbers is -10**

Example 6 The sum of two numbers is Negative three times the first number minus the second number equals 2. Find the numbers. Example 7 Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers. Assign labels. Translate each sentence. Solve the system. Write a sentence to give the answer.

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**Example 6 The sum of two numbers is -10**

Example 6 The sum of two numbers is Negative three times the first number minus the second number equals 2. Find the numbers. Let x = first number Let y = second number Translate each sentence. Solve the system. Write a sentence to give the answer. The numbers are 4 and -14.

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**Example 7 Four times one number minus three times another number is 12**

Example 7 Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers. Let x = first number Let y = second number Translate each sentence. Solve the system. Write a sentence to give the answer. The numbers are 3 and 0.

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Homework 5-A5 Page # 7–22,33-34,42-45.

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