solving systems of equations 5x - 2y = -24

Question

Did I solve these correctly?

 

1.  5x - 2y = -24

         -  4y = -48 - 10x

 

=   5x - 2y = -24

   10x  - 4y = -48

 

=  5x - 2y = -24 (multiply by -2)

=  -10x + 4y = 48

 

= -10x +4 y = 48

    10x  -4 y  =-48

 

    0  =  0

 

**If I get zero for both sides can I say this system cannot be solved?

 

 

2.  4x + 5y = -5

     3x +  y  = 21

 

I got x = 10 and y = 7.  When I check both equations then they do not equal.

 

**check first equation

 

4(10) + 5(7) = -5

     75 = -5     does not equal

 

3(10) + 7 = 21

    37 = 21     does not equal

 

How do I explain the system cannot be solved?

 

Thanks

 

 

 

 

 

 

Answer 1

Hi L;

(1)

I believe both equations are the same.  If I understand your information, the result is not zero but rather infinity.

 

2.

I will try to solve it...

4x + 5y = -5
3x + y = 21

Let's take the second equation and multiply both sides by 5...

15x+5y=105

Let's subtract this from the other equation...

4x + 5y = -5

-(15x+5y=105)

-11x+0=-110

-11x=-110

x=10

Let's plug this into either original equation.

I randomly select the first...

4x + 5y = -5

[(4)(10)]+5y=-5

40+5y=-5

y=-45

y=-9

Let's take the second original equation and plug-in x=10 and y=-9...

3x + y = 21

[(3)(10)]+-9=21

30-9=21

21=21

You were half-correct.

 

 

Answer 2

"Did I solve these correctly?

1. 5x - 2y = -24
- 4y = -48 - 10x

= 5x - 2y = -24
10x - 4y = -48

= 5x - 2y = -24 (multiply by -2)
= -10x + 4y = 48

= -10x +4 y = 48
10x -4 y =-48

0 = 0

**If I get zero for both sides can I say this system cannot be solved?"

 

YES, YOU SOLVED IT PERFECTLY! But your conclusion about 0 = 0 is wrong.

 

5x - 2y = -24
- 4y = -48 - 10x

 

Solving for y in both equations:

 

y = (5x + 24)/2

y = (10x + 48)/4 = (5x + 24)/2

 

As Vivian wrote, the two equations are the same.

 

The 0 = 0 indicates same equation because 0 = 0 is true for any x. Since every point on the line satisfies both original equations, there are an infinite number of solutions.

 

BTW, whenever you come up with something that is false for all x like 5 = 9, then you have parallel lines with no solutions.

 

Vivian's solution to #2 by elimination is correct.

 

You can also do it by substitution:

 

4x + 5y = -5
3x + y = 21

 

Solve for y in both equations:

 

y = (-4/5)x - 1

y = -3x + 21

 

Substitute -3x + 21 for y in 1st equation:

 

-3x + 21 = (-4/5)x - 1

 

Multiply both sides by 5:

 

-15x + 105 = -4x - 5

 

Add 15x + 5 to both sides:

 

110 = 11x

 

Divide by 11:

 

x = 10

 

y = -3(10) + 21 = -30 + 21

 

y = –9

 

Solution is (10,-9)