Help please

Question

I have to express each equation in slope-intercept form. Without graphing the equation, state whether the system of equations has exactly one solution, no solution, or an infinitive number of solutions.

 

x-6y=3

-4x+24y=-12

Answer 1

To solve this, we want to combine the equations. We can do this by first solving the first equation for x:

 

x-6y=3, so if we add 6y to both sides, we are left with:

x=3+6y

 

To make this easier, we can simplify the second equation by dividing all components by 4.

 

-4x+24y=-12

(-4x)/4 + (24y)/4 = (-12)/4

 

We are left with this for our equations:

x=3+6y (after solving the first equation for x)

and

-x + 6y = -3 (after reducing the second equation by 4)

 

To combine these, let's substitute the first equation for x in the second equation. 

-(3+6y) + 6y = -3

 

Remember to distribute the negative:

-3 -6y + 6y = -3

 

Combine like terms:

-6y + 6y = 0

0 = 0

 

Because 0=0 for this system (and 0 will always equal 0 in the world of mathematics), we know that all solution(s) of these equations will be true. Notice that I said all solution(s), not all real numbers. 

 

To check this, let's solve for x in both equations and compare them. We have already solved for x in the first equation and were left with x=3+6y. For the second, we can see:

 

-4x+24y=-12
(-4x)/4 + (24y)/4 = (-12)/4

-x + 6y = -3

x - 6y = 3

x = 3 + 6y

 

Because these equations are the same when solved for x, we know that there are infinite solutions. 

Answer 2

Hi Hillary from Clinton,

 

The solution for a system of two linear equations is the point(s) they have in common. There are three options:

1.) If the two lines intersect, then the solution is the ordered pair where they intersect.

2.) If the two lines are coincidental, or are the same line, think one line on top of one another, they intersect at every point on the lines. This system has an infinite number of solutions.

3.) If the lines are parallel, they never touch, so there is no solution to the system.

 

We will need to put both of your equations into slope intercept form (solve for y).

 

y= mx + b

 

If the slopes (m), are different, the lines have to intersect.

 

If the slopes (m), are the same AND the y-intercepts (b) are the same, then the lines are coincidental.

 

If the slopes (m), are the same, but the y-intercepts (b) are different, then the lines are parallel.

 

Sound good?

 

For example:

y = 2x + 5
y = 2x + 9

The equation for the slope-intercept form is y = mx + b. In both of the equations above, they have the SAME slope (m) but they have DIFFERENT y-intercepts (b), so they are parallel, they never intersect, and there is no solution.

 

Ready to solve your equations for y?

 

x - 6y = 3

  -6y = -x + 3

     y = (1/6)x - (1/2)

 

-4x + 24y = -12

         24y = 4x - 12

            y = (1/6)x - (1/2)

 

Notice, the slopes are the SAME (m = 1/6) and the y-intercepts are the SAME (b = -1/2). Therefore, they are coincidental, the same line, they intersect at every point on the lines, and there is an infinite number of solutions.

 

I hope this helps!