Question

Which of the equations have no solutions with the equation below?

−2x+6y=6
4x−y=−12
−15+x=3y
5x−8y=−10
x−y=2

Answer 1

To determine which equations have no solutions with the equation \(-2x + 6y = 6\), we first convert this equation into slope-intercept form (i.e., \(y = mx + b\)) to identify its slope and y-intercept.

Starting with the equation:

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

Add \(2x\) to both sides:

\[
6y = 2x + 6
\]

Now, divide everything by 6:

\[
y = \frac{1}{3}x + 1
\]

Now we know the slope is \(\frac{1}{3}\) and the y-intercept is \(1\). Any line that has the same slope and a different y-intercept will be parallel to this line, and therefore, those lines will not intersect and thus have no solutions.

Now let's analyze each equation and convert them to slope-intercept form to find their slopes.

1. **\(4x - y = -12\)**

Rearranging gives:

\[
-y = -4x - 12 \quad \Rightarrow \quad y = 4x + 12
\]

Slope = 4

2. **\(-15 + x = 3y\)**

Rearranging gives:

\[
x + 15 = 3y \quad \Rightarrow \quad 3y = x + 15 \quad \Rightarrow \quad y = \frac{1}{3}x + 5
\]

Slope = \(\frac{1}{3}\) (this is the same as our original equation but with a different y-intercept).

3. **\(5x - 8y = -10\)**

Rearranging gives:

\[
-8y = -5x - 10 \quad \Rightarrow \quad y = \frac{5}{8}x + \frac{5}{4}
\]

Slope = \(\frac{5}{8}\)

4. **\(x - y = 2\)**

Rearranging gives:

\[
-y = -x + 2 \quad \Rightarrow \quad y = x - 2
\]

Slope = 1

Now we can summarize the slopes:

- \(-2x + 6y = 6\) — Slope: \(\frac{1}{3}\)
- \(4x - y = -12\) — Slope: 4
- \(-15 + x = 3y\) — Slope: \(\frac{1}{3}\) (same slope, different y-intercept)
- \(5x - 8y = -10\) — Slope: \(\frac{5}{8}\)
- \(x - y = 2\) — Slope: 1

The equations that have the same slope as \(-2x + 6y = 6\) are:

- \(-15 + x = 3y\)

Thus, the equation that has **no solutions** with \(-2x + 6y = 6\) is:

\[
-15 + x = 3y
\]

This is the only equation among the listed equations that is parallel to \(-2x + 6y = 6\).