For what value of the constant (c) does the system of equations below have no solutions (x,y)?

3x - 5y = -2.3
6x - cy + 9.3

How would I start to figure this out? Thank you!

Well, well, well, looks like we have a sneaky little system of equations here! To find the value of the constant (c) for which the system has no solutions, we need to see if the two equations are parallel or not.

Let's start by putting the second equation in proper form. It seems like our constant (c) escaped from the equation, so let's wrangle it back in:

6x - cy = -9.3

Now, compare the coefficients of x and y in the two equations. We have 3 and -5 in the first equation, and 6 (the coefficient of x) and -c (the coefficient of y) in the second equation.

If the ratio of the coefficients of x is the same as the ratio of the coefficients of y, then we can say that the two lines are parallel. So, to find the value of (c) for which the system has no solutions, we need to make sure the ratio of the coefficients is NOT the same.

In this case, the ratio of the coefficients of x is 3/6 = 1/2.

To be funky, let's see if the ratio of the coefficients of y is also 1/2. If we set the ratio of the coefficients equal to each other, we get:

-5/(1/2) = -c/1

Simplifying this equation, we get:

-10 = -c

Voila! We have found the value of the constant (c) for which the system has no solutions. The answer is c = 10!

So, my dear friend, when c equals 10, the system of equations will have no solutions (x, y). Keep those equations in check and always have a sense of humor while solving math problems!

What would happen to the graphs of the the two lines be if c were 10?

Think slope.

Clown Bot, how did you get -5/(1/2)=-c/1?

To find the value of the constant (c) that makes the system of equations have no solutions for (x, y), we need to determine when the two equations represent parallel lines.

First, let's rewrite the second equation in the standard form (Ax + By = C), where A, B, and C are constants:

6x - cy = -9.3

Now, let's compare the coefficients of x and y in both equations:

Equation 1: 3x - 5y = -2.3
Coefficient of x: 3
Coefficient of y: -5

Equation 2: 6x - cy = -9.3
Coefficient of x: 6
Coefficient of y: -c

For the two lines to be parallel, the ratios of the coefficients of x and y need to be equal. In other words, the slope of one line must be equal to the slope of the other line.

So, to find the value of c, we need to set the ratio of the coefficients of y in both equations equal to each other:

-5 = -c

Solve the equation:

c = 5

Therefore, the value of the constant (c) that makes the system of equations have no solutions is c = 5.