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Systems of Equations Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 4, 2011

Elimination by Addition Method (Combination Method)

Solving a system of equations by substitution can be messy when none of the coefficients is 1. Fortunately, there is another way. We can always add the two equations to eliminate one of the variables. Sometimes, though, we need to multiply one or both equations by a number to make it work.

Systems of Equations and Inequalities Examples

Fig. 10.3

Example

Solve the systems of equations. Put your solutions in the form of a point, (x, y) .

  • Systems of Equations and Inequalities Example
  • Add the equations by adding like terms. Because we will be adding −3 y to 3 y , the y -term will cancel, leaving one equation with only one variable.

Systems of Equations and Inequalities Example

We can put x = 2 into either A or B to find y . We will put x = 2 into A.

2 x − 3 y = 16

2(2) − 3 y = 16

−3 y = 12

y = −4

The solution is (2, −4).

Multiplying to Cancel Variables

Sometimes we need to multiply one or both equations by some number or numbers so that one of the variables cancels. Multiplying both sides of any equation by a nonzero number never changes the solution.

Examples

  • Systems of Equations and Inequalities Examples
  • Because the coefficients on y are the same, we only need to make one of them negative. Multiply either A or B by −1, then add.

Systems of Equations and Inequalities Examples

The solution is (2, −3).

 

  • Systems of Equations and Inequalities Examples

Several options will work. We could multiply A by −2 so that we could add −4 x (in −2A) to 4 x in B. We could multiply A by 2 and multiply B by 7 so that we could add 14 y (in 2A) to −14 y (in 7B). We could also divide B by −2 so that we could add 2 x (in A) to −2 x (in − Systems of Equations and Inequalities Examples B). We will add −2A + B.

Systems of Equations and Inequalities Examples

The solution is (4, −1).

Both equations in each of the following systems will need to be changed to eliminate one of the variables.

Examples

  • Systems of Equations and Inequalities Examples
  • There are many options. Some are 3A − 8B, −3A + 8B, and 2A + 5B. We will compute 2A + 5B.

Systems of Equations and Inequalities Examples

The solution is (1, 2).

 

  • Systems of Equations and Inequalities Examples
  • First, we will eliminate the fractions. The LCD for A is 72, and the LCD for B is 30.

48 x − 18 y = 25 72A

15 x + 12 y = −1 30B

Now we will multiply the first equation by 2 and the second by 3.

Systems of Equations and Inequalities Examples

Practice problems for this concept can be found at: Systems of Equations and Inequalities Practice Test.

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