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Elimination Using Addition and Subtraction

Elimination Using Addition and Subtraction. Adding Opposites. Previously, we learned that adding two numbers together which have the same absolute value but are opposite in sign results in a value of zero. This can be shown graphically using algebra tiles.

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Elimination Using Addition and Subtraction

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  1. Elimination Using Addition and Subtraction

  2. Adding Opposites Previously, we learned that adding two numbers together which have the same absolute value but are opposite in sign results in a value of zero. This can be shown graphically using algebra tiles. Each algebra tile has the same absolute value but is opposite in sign. Each pair of tiles makes a zero pair. This same principle can be applied to variables. Below is a zero pair for the variable, “x”.

  3. Zero pair Elimination We can apply the idea of a zero pair to systems of equations so that one of the variables can be eliminated. Look at the following system of equations. What do you observe? In each equation, there is a “y”. However, the signs in the two equations are opposite. If added together, the “y” and the “-y” would make a zero pair.

  4. Addition Add the following two equations together by adding like terms together. Now pick one of the two equations and substitute the value of “x” into that equation and solve for “y”. x + y = 6 4 + y = 6 -4 -4 Now solve for “x”. y = 2 2x = 8 Write the solution as an ordered pair. 2 2 x = 4 Solution: (4,2)

  5. Addition Solution: (4,2) To check your work, substitute the values for the variables into each equation and determine if it is true.   The solution is correct.

  6. Subtraction Addition worked when the signs of one of the variables were opposite. However, you may encounter a system of equations in which the signs of the variables are the same. In this case, instead of adding, you will be subtracting one equation from the other. Remember that subtraction is really the same as adding the opposite. Let’s take a look at the following system of equations: The signs of the variables are all positive. Therefore, in order to solve this system, we can subtract the bottom equation from the top equation.

  7. Subtraction Since we have the value for “x”. Pick one of the equations and substitute the value 3 for “x” and solve for “y”. Subtract the bottom equation from the top equation. 3x + y = 6 (3)(3) + y = 6 9 + y = 6 -9 -9 (-) or y = -3 Write the solution as an ordered pair. Solution: (3,-3)

  8. Subtraction Solution: (3,-3) To check your work, substitute the values for the variables into each equation and determine if it is true.   The solution is correct.

  9. No Solution Solve the following system of equations by using addition or subtraction. When subtraction is used, there are no more variables (x or y) remaining and the result is an incorrect statement. We know that This system of equations can be solved by using subtraction. Therefore, this system of equations represents parallel lines and there is no solution. (-)

  10. Summary of Steps • Arrange the two equations so that the like terms are in vertical columns. • If the signs of the variables are opposite, add the two equations together to eliminate one of the variables. • If the signs of the variables are the same, then subtract one of the equations from the other equation. • Solve for the remaining variable. • Substitute the value of the variable into one of the equations and solve for the other variable. • Check the solution by substituting the values of the two variables into each equation. • Write the solution. If there is one solution, write it as an ordered pair.

  11. You Try It Solve each system of equations by using addition or subtraction. • The sum of two numbers is 85. The difference between • the two numbers is 19. What are the two numbers?

  12. Problem 1 Add the two equations together. Solve for “y” by substituting the value for “x” into one of the equations. (+) Solve for “x”. 3x = 21 3 3 x = 7 Solution: (7,1)

  13. Problem 2 Solve for “s” by substituting the value for “r” into one of the equations. Subtract the bottom equation from the top equation. (-) Solve for “r”. Solution: (5,-5)

  14. Problem 3 The sum of two numbers is 85. The difference between the two numbers is 19. What are the two numbers? Write an equation for the sum of the numbers. Write an equation for the difference of the numbers. Add the equations together. Solve for x.

  15. Problem 3 Solve for “y” by substituting in the value of “x”. The two numbers are 52 and 33.

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