Solve the system, or show that it has no solution

Question

Solve the system, or show that it has no solution. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, enter the general solution in terms of x, where x is any real number.) 
−4x
 + 
12y
 = 
4
12x
 + 
4y
 = 
188
(x, y) = 
 

Answer 1

-4X+12Y=4
12X+4Y=188

 

The equations have slopes that a negative reciprocals. The lines are perpendicular. 

There is one solution.

 

-12x + 36y = 12       multiply both sides by 3

12x + 4y = 188

 

          40y = 200

 

What is the solution?

Answer 2

(1) -4x + 12y = 4

(2) 12x + 4y = 188

 

Our goal is by combining the two equations, we can get rid of one of the variables.

By multiply both sides of equation (1) by 3, we will wind up with -12x in

a new version of the first equation and 12x in the second.

 

    -12x + 36y = 12

 

add this to equation (2)

  12x +  4y = 188

-12x + 36y =   12

------   ----    -----

           40y = 200

therefore y = 5.

 

now plug 5 in for the value of y in one of the first two equations. You will end up with an equation that only has x's and constants.

 

I will use equation (2).

 

12x + 4*5 = 188

12x + 20 = 188

12X = 168            (by subtracting 20 from both sides)

x = 14                ( divide both sides by 12)

 

So, the only value of y that satisfies both equations is 50 and the only value of x is -1.

(x,y) = (14, 50)

 

[you should check this by substituting these values into the two equations to make sure that you did your work correctly.]

 

[If you wanted to check it a second way, you could graph the two functions and see where they cross.]

[if they are parallel lines, there are no solutions, if they are co-linear (they are on top of one another), there are an infinite number of solutions]

Answer 3

You can solve this system by substitution, elimination, or graphing.  To solve by elimination:

 

1. Multiply one or both expressions by a constant to eliminate one variable.

 

[-4x + 12y = 4] 3

12x + 4y = 188

 

-12x + 36y = 12

12x + 4y = 188

 

2. The variable x is eliminated, so add the remaining variables and constants.

 

40y = 200

y = 5

 

3. Plug in the value for y into one of the original equations to find x.

 

-4x + 12y = 4

-4x + 12(5) = 4

-4x + 60 = 4

-4x = -56

x = 14

 

The solution to the system of equations is (14,5).