Try rewriting each equation in the form $y = mx + b$ so you can clearly see the slope of each line. What part of this form represents the slope?

Once you have both slopes, set them equal to find the value of $p$ that makes the lines parallel. After that, check that the y-intercepts are not equal, so the lines are different and there is no solution.

In one expression line, type (p+3)x - 4y = 8. Desmos will create a slider for p. In another expression line, type 5x - 20y = 42 to graph the second line.

Use the p-slider to set $p$ equal to each answer choice in turn: $-7$, $-2$, $2$, and $7$. For each value, look at the graph to see whether the two lines cross in a single point, overlap completely, or stay separate.

The correct value of $p$ is the one where the two lines are parallel (same steepness and direction) and do not intersect anywhere on the graph. Note which answer choice produced that situation.

For a system of two linear equations, having no solution means there is no point $(x,y)$ that satisfies both equations at the same time.

On a graph, this happens when the two lines are parallel but not the same line:

• same slope
• different y-intercepts (so they never meet).

Rewrite each equation in slope-intercept form $y = mx + b$.

First equation:

Move the $x$-term to the right:

Divide both sides by $-4$:

So the slope of the first line is $\dfrac{p+3}{4}$.

Second equation:

Move $5x$ to the right:

Divide both sides by $-20$:

So the slope of the second line is $\dfrac{1}{4}$.

For the lines to be parallel (same steepness and direction), their slopes must be equal.

Set the slopes equal:

Clear the denominator by multiplying both sides by $4$:

Now solve this equation to find the value of $p$ that makes the two lines parallel.

Solve $p + 3 = 1$:

Now check that the lines are different (so there is no solution, not infinitely many solutions).

When $p = -2$, the first equation becomes

In slope-intercept form, this is

The second line is

which has the same slope $\dfrac{1}{4}$ but a different y-intercept.

So the lines are parallel and never intersect, meaning the system has no solution when $p = -2$. This corresponds to answer choice B.