Solving equation question

Question

A system of equations is shown below. What is the value of X in the solution to the system?

 

y = 1.5x - 2.5

2x - 6y = -6

Answer 1

Hi Latoya,

The problem presents two equations of intersecting lines:

       y = 1.5x - 2.5                this form is called “slope-intercept” form
        2x - 6y = -6                  this form is called “standard” form

Many of the problems you will have will use one or both of these forms.  Since they are not parallel, they intersect at exactly one point.  Finding that point is called “solving” the system of equations.  We usually express the solution in (x,y) form.   [note: for this x(x,y)  both equations are true and we will check that at the end.

There are two methods that you will learn for solving such equations – (1) substitution and (2) elimination.  Learn them both, then you may chose the easiest/fastest when you see this type of problem on a test (unless the test question specifies which method to use).

Using substitution:
     Put the value of y (from first equation)  in place of every y in the second equation:
          2x  - 6(1.5x - 2.5) = -6
             -7x  + 15  = -6
                     -7x =  -21
                        x=3

              and put that x into either equation to solve for y:
                2(3) -6y = -6
                   -6y = -12
                     y = 2

Or, by elimination (get the same coefficient of one of the terms so it will become 0 when we add equations):
       -1.5x + y  = -2.5
          2x  - 6y = -6

    Choose one.  Let’s get rid of the y by multiplying the first equation by 6 (you will learn to find the easiest)
             -9x  +  6y   =   -15
              2x   -  6y   =    -6
         -----------------------------------   (add the two equations)
             -7x   +  0    =   -21     
                   x = 3

                 and (putting x into either equation)
           2(3) – 6y  =  -6
                    -6y = -12
                       y = 2

Checking:
      Is  2 = 1.5(3) -2.5    ?
           2  =  4.5 – 2.5      Yes.
      Is  2(3) – 6(2) = -6   ?
           6 – 12   =   -6       Yes.

 

The solution is (3,2).  Since the question specifically asked for the value of x in the solution, the answer is x=2.

 

Answer 2

y = 1.5x - 2.5              eq1
2x - 6y = -6                 eq2

 

Arthur showed you how to solve these equations by using the elimination method which places emphasis on multiplying and adding/subtracting equations.

 

We can also use the substitution method, since y is already defined by eq1.  Substitute eq1 into eq2.

 

2x - 6(1.5x - 2.5) = -6

 

 

We now have a single equation with only one variable to work with. Solve for x.

 

2x - 9x + 15 = -6

-7x + 15 = -6

-7x = -21

   x = 3

 

Substitute this value of x into eq1 to solve for y.

 

y = 1.5(3) - 2.5

y = 4.5 - 2.5

y = 2

 

Your solutions are

 

x = 3

y = 2

 

Answer 3

y=1.5x-2.5

2x-6y=-6

rearrange terms in the first equation

-1.5x+y=-2.5

2x-6y=-6

multiply the first equation by 6

-9x+6y=-15

2x-6y=-6

add

-7x=-21

divide both sides by -7

x=3

substitute into either equation

2*3-6y=-6

6-6y=-6

-6y=-12

y=2

(3,2) is the solution