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

For a system of equations to have an infinite number of solutions, the two equations must represent the same line. This means that the second equation must be a scalar multiple of the first equation.

Let's start with the first equation:

\[ 6x + 3y = 9. \]

First, we can convert this equation into slope-intercept form (y = mx + b).

1. Subtract \( 6x \) from both sides to isolate the \( y \) term:

\[ 3y = -6x + 9 \]

2. Divide every term by 3:

\[ y = -2x + 3. \]

Now that we have it in slope-intercept form, we can create a second equation that is equivalent to the first by multiplying the entire equation by a non-zero constant. Let's choose a multiplier of \( 2 \):

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

Distributing the \( 2 \):

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

Now, rearranging this into slope-intercept form, we can write it as:

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

or by dividing by \( 2 \):

\[ y = -2x + 3. \]

Since this is equivalent to the first equation, you can also multiply it by \( -1 \):

This gives us another form:

\[ 2y + 4x = 6, \]

To provide a different valid form, let's also multiply our original by \( \frac{1}{2} \):

This leads us to:

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

So, any equation that is a scalar multiple of \( 6x + 3y = 9 \) will be valid. Another example could be something along the lines of:

\[
4y = -8x + 12.
\]

Including multiples such as:

- \( 9x + 6y = 27 \)
- Many such equations yield infinite solutions.

Thus, you may say explicitly:

A possible second equation is:

\[ 6x + 3y = 9. \]

But to be completely clear, a different scalar multiple leading to a newly represented equation could be:

\[
y + 2x = 3.
\]

This will preserve the properties of there being infinitely many solutions.