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Solving Systems of Equations Algebraically Study Guide

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Updated on Aug 24, 2011

Find practice problems and solutions for these concepts at Solving Systems of Equations Algebraically Practice Problems.

You know how to solve systems of equations graphically. In this lesson, you will learn how to solve systems of equations algebraically. You will learn how to use the elimination method and the substitution method for solving systems of equations.

How to Use the Elimination Method

Graphs serve many useful purposes, but using algebra to solve a system of equations can be faster and more accurate than graphing a system. A system of equations contains equations with more than one variable. If you have more than one variable, you need more than one equation to solve for the variables. When you use the elimination method of solving equations, the strategy is to eliminate all the variables except one. When you have only one variable left in the equation, then you can solve it.

Example:

x + y = 10

xy = 4

Add the equations. (x + y) + (xy) = 10 + 4
  2x – 0y = 14
Drop the 0y. 2x = 14
Divide both sides of the equation by 2. Solving Systems of Equations Algebraically
Simplify both sides of the equation. x = 7

 

You have solved for the variable x. To solve for the variable y, substitute the value of the x variable into one of the original equations. It does not matter which equation you use.

  x + y = 10
Substitute 7 in place of the x variable. 7 + y= 10
Subtract 7 from both sides of the equation. 7 – 7 + y = 10 –7
Simplify both sides of the equation. y = 3

 

You solve a system of equations by finding the value of all the variables. In the previous example, you found that x = 7 and y = 3. Write your answer as the ordered pair (7,3). To check your solution, substitute the values for x and y into both equations.

Check:  x + y= 10

Substitute the values of the variables into the first equation. 7 + 3 = 10
Simplify. 10 = 10
  xy = 4
Substitute the values of the variables into the second equation. 7 – 3 = 4
Simplify. 4 = 4

 

Did you get the right answer? Because you got true statements when you substituted the value of the variables into both equations, you solved the system of equations correctly. Try another example.

Example:

x + y = 6

x + y = –4

Add the two equations. 0x + 2y = 2
Drop the 0x. 2y = 2
Divide both sides of the equation by 2. Solving Systems of Equations Algebraically
Simplify both sides of the equation. y = 1
Use one of the original equations to solve for x. x + y = 6
Substitute 1 in place of y. x + 1 = 6
Subtract 1 from both sides of the equation. x + 1 – 1 = 6 – 1
Simplify both sides of the equation. x = 5

Write the solution of the system as an ordered pair: (5,1).

Check: x + y = 6

Substitute the values of x and y into the first equation. 5 + 1 = 6
Simplify. 6 = 6
  x + y = –4
Substitute the values of x and y into the second equation. –(5) + 1 = –4
Simplify. –4 = –4

Did you get the right answer? Yes! You got true statements when you substituted the value of the variables into both equations, so you can be confident that you solved the system of equations correctly.

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