Question

-3x + 4y = -6

Answer 1

There is no any c values that would produce a system with no solution because the only c value that makes the line 2 to have the same slope as line 1 also produces the same y-intercept.

Explanation:

Part A.

A system that has an infinite number of solutions is called Consistent and dependent. If we have a system of linear equations in two variables, then this system will have an infinite number of solutions if and only if the two equations are basically the same. In this case, we have the following system:

$\left \{ {{-3x + 4y = -6} \atop {cx + 8y = -12}} \right.$

Notice that if we divide the coefficient of the variable y we get:

$\frac{8}{4}=2$

And if we divide the constant terms we get:

$\frac{-12}{-6}=2$

So, in order to get c we have:

$\frac{c}{-3}=2 \\ \\ \\ Isolating \ c: \\ \\ c=-3(2) \\ \\ c=-6$

Part b.

A system with no solution is called inconsistent in whose case the two equations will have the same slope but different y-intercepts.

For the first equation we have:

$-3x + 4y = -6$

But, isolating y, we can write this in slope intercept form :

$y=-\frac{3}{4}x-\frac{3}{2}$

For the second equation we have:

$cx + 8y = -12$

But, isolating y, we can write this in slope intercept form :

$y=-\frac{c}{8}x-\frac{3}{2}$

So the only possible for the line to have the same slope as 1 is that:

$-\frac{c}{8}=-\frac{3}{4} \\ \\ c=8(\frac{3}{4}) \\ \\ c=6$

So they have the same slope but they also have the same y-intercept making the lines to be the same.

In conclusion, there is no any c values that would produce a system with no solution because the only c value that makes the line 2 to have the same slope as line 1 also produces the same y-intercept.

Learn more:

Solving system of linear equations:

brainly.com/question/13799715

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