For equations written as $y=mx+b$, compare the slopes $m$. How does having the same slope versus different slopes affect the number of intersection points?

Equations like $x=4$ are vertical lines—how do two different vertical lines interact? For equations not already in $y=mx+b$ form, check whether one equation is just a constant multiple of the other.

In Desmos, enter y = 3x - 5 and y = 3x + 4 as two separate lines. Observe whether these lines ever cross or whether they stay the same distance apart (parallel) everywhere.

Enter x = 4 and x = 10. These graph as vertical lines. Check visually if they intersect at any point on the coordinate plane.

Enter 2x + 4y = 8 and x + 2y = 4. Notice whether the two graphs lie exactly on top of each other or appear as two distinct lines. If they overlap perfectly, that means the system has infinitely many solutions.

Enter y = -2x + 1 and y = 4x - 3. Look at how these two lines relate: do they intersect, and if so, how many intersection points do you see? Compare the number of intersections here with what you observed for the other options and with the question’s requirement of “exactly one solution.”

Each linear equation in two variables graphs as a straight line.

For two lines:

• If they intersect once, the system has exactly one solution.
• If they are parallel and distinct, they never intersect ⇒ no solution.
• If they are the same line, they intersect at infinitely many points ⇒ infinitely many solutions.

In slope-intercept form $y=mx+b$:

• Same slope $m$, different $b$ ⇒ parallel, no solution.
• Same slope $m$, same $b$ ⇒ same line, infinitely many solutions.
• Different slopes ⇒ exactly one solution.

Both equations are already in the form $y=mx+b$.

• First line: slope $m=3$, intercept $b=-5$.
• Second line: slope $m=3$, intercept $b=4$.

The slopes are the same and the intercepts are different, so these are parallel lines that never intersect.

This system has no solution, so it does not have exactly one solution.

Equations of the form $x=\text{constant}$ are vertical lines.

• $x=4$ is a vertical line through $x=4$.
• $x=10$ is a vertical line through $x=10$.

Both are vertical, so they are parallel vertical lines. Since they are different lines, they never intersect.

This system also has no solution, so it does not have exactly one solution.

Look at how the two equations relate:

• If you multiply the second equation by 2, you get

This is exactly the first equation. That means both equations represent the same line.

When two equations describe the same line, every point on that line is a solution to the system.

So this system has infinitely many solutions, not exactly one.

Now look at the last pair, which is also in the form $y=mx+b$:

• First line: slope $m=-2$, intercept $b=1$.
• Second line: slope $m=4$, intercept $b=-3$.

Here, the slopes are different ($-2$ and $4$). Lines with different slopes are not parallel, so they must intersect at exactly one point.

Therefore, the system $y = -2x + 1$ and $y = 4x - 3$ is the only system given that has exactly one solution.