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)

Answer 1

To have an infinite number of solutions, the second equation must be equivalent to the first equation. This means it should be a multiple of the original equation.

Given the equation:
\[ 6x + 3y = 9 \]

We can rearrange this into slope-intercept form (y = mx + b):

1. Subtract \( 6x \) from both sides:
\[ 3y = -6x + 9 \]

2. Divide everything by 3:
\[ y = -2x + 3 \]

Now, to find an equivalent equation, we can multiply the entire equation by any non-zero scalar. Let's say we multiply it by 2:

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

Rearranging it back to standard form gives us:
\[
4x + 2y = 6
\]

So, an example of a second equation that has an infinite number of solutions with the first equation is:

\[
4x + 2y = 6
\]

Alternatively, we could also write it in slope-intercept form. Let's say we just want to double the original equation, it would lead us to:

\[
y = -2x + 3 \quad \text{(which is the same as the first equation in slope-intercept form)}
\]

Thus, the equation in slope-intercept form could be:
\[
y = -2x + 3
\]