system of equations

Question

Explain, in complete sentences, how you would use the elimination method to solve the following system of equations and provide the solution to the system.

5x - 9y = -16
2x + 6y = -16

Answer 1

May, you'll want to look for the least common factors of each variable.

 

The least common factor of the x coefficients being 2 and 5 is 10.

The least common factor of the y coefficients being 9 and 6 is 36.

 

Hence, the calculations would become easier if you eliminate the x variable.

 

Multiply the first equation by 2 in which becomes 10x-18y=-32.

Multiply the second equation by -5 in which becomes -10x-30y=80

 

Add the two equations in which the final equation becomes -48y=48

 

Divide both sides of the equation by -48. This makes y=-1.

 

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

 

Lets substitute y=-1 into the first equation in which it becomes 5x-9(-1)=-16

 

Remove the parenthesis in which the equation becomes 5x+9=-16

 

Subtract 9 from both sides of the equation in which the eqaution becomes 5x=-25

 

Divide both sides of the equation by 5 in which makes x=-5.

 

Therefore the solution to the system is (-5,-1).

Answer 2

Hi May

 

First we need to eliminate the same variable from the equations. We will eliminate the x-variable by multiplying the second equation by -5/2. We multiply each term, the 2x, 6y and -16 by -5/2.

 

This will result in the second line now being -5x - 15y = 40.

 

5x - 9y = -16

-5x -15y = 40

 

Now we ELIMINATE the x-variable by adding straight down from line I to line II.

We have 5x + (-5x) equaling zero (0).

We have -9y + (-15y) equaling -24y.

We have -16 + 40 equaling 24.

Now we have the single equation -24y = 24.

 

We now divide each side of the equation by -24 to get positive y isolated and equal to a single value.

 

When we divide each side of the equation by -24, we get y = -1.

 

Now we need to substitute the value for y into one of the original equations (Line I or Line II).

 

Substituting y = -1 into Line I gives us the equation:

 

5x - (9)(-1) = -16

 

We multiply 9 and -1 to get -9.

 

Our equation now reads:

 

5x - (-9) = -16  ----> This simplifies to be 5x + 9 = -16.

 

We now subtract 9 from each side of the equation.

 

This gives us:

 

5x = -25.

 

We divide each side of the equation by 5 to isolate the x-variable.

 

By doing so, we get x = -5.

 

So we have x = -5 and y = -1.

 

We check our solutions by substituting both the value for x and the value for y into either Line I or Line II.

 

Substituting the x and y values into Line II yields:

 

2(-5) + 6(-1) = -16

 

Is this statement correct?

 

-10 + (-6) = -16

 

YES.

 

Therefore, x = -5 and y = -1 by means of the elimination method.

Answer 3

Hello there, May

 

Hey, I love to explain answers in detail.. so you got the right person for this problem :).. Here we go

 

There is an infinite number of ways to do this problem.  We have to decide which variable is the easiest to get rid of, the ones that require the least amount of calculations.  We can eliminate the x by making one into 10x and the other into -10x.  Or we can eliminate the y by making one into 18y and the other into -18y.  I am going to do the latter operation.  I am going to eliminate the y first.  Again, it is always up to you what to eliminate.. keep it simple, and try to avoid dealing with negatives.

 

5x-9y=-16       Multiply both sides by 2 in order to make -9y into -18y

2x+6y=-16      Multiply both sides by 3 in order to make 6y into 18 y

 

2(5x-9y=-16)    Using the distributive property we get

10x-18y=-32

 

3(2x+6y=-16)    Using the distributive property we get

6x+18y= -48

 

So, we are ready to get rid of the y.. combine (add) to top equation to the bottom equation

 

10x-18y=-32

6x+18y=-48       The y's will cancel each other out, so we get

 

16x=-80             Divide by 16

x=-5

 

What we must do now, May, is to substitute -5 for x into either of the original equations.  Let's use the second equation.

 

2x+9y=-16       Substituting -5 for x

2(-5)+6y=-16   

-10+6y=-16     Add 10 to both sides

+10       +10

      6y=-6        Divide both sides by 6

       y=-1

So, our answer is (-5, -1).. Next, we proceed to check our answer by plugging -5 for x and -1 for y into the original equations.

 

5x-9y= -16      Substituting

5(-5)-9(-1)=-16

-25+9=-16

-16=-16         That's true.. So, our answer is correct.

 

Ok, May, it was a pleasure to help out.. If you are still confuse, just read it a few times carefully, I assure you that you will get it.  Take care, and please rate my answer. Thanks.. :)

 

D. Y. Taylor