Question

elimination and substitution method example

Answer 1

if it helps you mark as brainlieast.

Answer 2

The elimination method for solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an equation.

 

So if you have a system: x – 6 = −6 and x + y = 8, you can add x + y to the left side of the first equation and add 8 to the right side of the equation. And since x + y = 8, you are adding the same value to each side of the first equation.

 

Using Addition to Eliminate a Variable

If you add the two equations, x – y = −6 and x + y = 8 together, as noted above, watch what happens.

 



 

You have eliminated the y term, and this equation can be solved using the methods for solving equations with one variable.

 

Let’s see how this system is solved using the elimination method.

 

 

Example

Problem

Use elimination to solve the system.

   x – y = −6

x + y = 8

 

 



Add the equations.

 

2x = 2

x = 1

Solve for x.

 

x + y = 8

1 + y = 8

y = 8 – 1

y = 7

Substitute x = 1 into one of the original equations and solve for y.

 

x – y = −6

1 – 7 = −6

−6 = −6

 

TRUE

x + y = 8

1 + 7 = 8

8 = 8

 

TRUE

Be sure to check your answer in both equations!

 

 

The answers check.

Answer

The solution is (1, 7).

 

 

 

Unfortunately not all systems work out this easily. How about a system like 2x + y = 12 and −3x + y = 2. If you add these two equations together, no variables are eliminated.

 



 

But you want to eliminate a variable. So let’s add the opposite of one of the equations to the other equation.

 

 2x + y =12      →        2x + y = 12      →       2x + y = 12

           −3x + y = 2      →      − (−3x + y) = −(2)   →  3x– y = −2

                                                                                    5x + 0y = 10

 

You have eliminated the y variable, and the problem can now be solved. See the example below.

 

 

Example

Problem

Use elimination to solve the system.

2x + y = 12

                                −3x + y =  2

 

 

2x + y = 12

            −3x + y =  2

You can eliminate the y-variable if you add the opposite of one of the equations to the other equation.

 

 2x + y = 12

            3x – y =  −2

5x = 10

Rewrite the second equation as its opposite.

Add.

 

x = 2

Solve for x.

 

2(2) + y = 12

4 + y = 12

y = 8

Substitute y = 2 into one of the original equations and solve for y.

 

2x + y = 12

2(2) + 8 = 12

4 + 8 = 12

12 = 12

 

            TRUE

−3x + y =  2

−3(2) + 8 = 2

−6 + 8 = 2

2 = 2

 

TRUE

Be sure to check your answer in both equations!

 

 

 

The answers check.

Answer

The solution is (2, 8).