Question

Cruz wrote a system of equations with an infinite number of solutions. One of his equations is 6x+3y=9
. What could be his second equation? (Write the equation in slope-intercept form.)(1 point)
The second equation could be y=

Answer 1

To create a second equation that has the same infinite solutions as the first equation \(6x + 3y = 9\), we need to ensure that the second equation is a multiple of the first equation.

First, we can simplify the first equation:

1. Divide the entire equation by 3:

\[
2x + y = 3
\]

2. Now, convert this equation to slope-intercept form \(y = mx + b\):

\[
y = -2x + 3
\]

Now, we can create a second equation by multiplying the entire equation by any non-zero constant \(k\). For example, let's choose \(k = 2\):

\[
2(2x + y) = 2(3)
\]

This leads to:

\[
4x + 2y = 6
\]

Now, we can convert it into slope-intercept form:

\[
2y = -4x + 6
\]
\[
y = -2x + 3
\]

We can choose any constant multiple. For instance, if \(k = 1/2\):

\[
\frac{1}{2}(6x + 3y) = \frac{1}{2}(9)
\]

This simplifies to:

\[
3x + \frac{3}{2}y = \frac{9}{2}
\]

Converting this equation to slope-intercept form gives:

\[
\frac{3}{2}y = -3x + \frac{9}{2}
\]
\[
y = -2x + 3
\]

Thus, one example for Cruz's second equation could be:

\[
y = -2x + 3
\]

Or he could choose any other multiple of the first equation, for example:

\[
y = -4x + 12
\]

So, a possible answer (which maintains an infinite number of solutions) could also be:

\[
y = -4x + 12
\]

You can express this as:

\[
y = -2x + 3
\]

as already mentioned, or in other equivalent forms.