Instead of graphing, use the coefficients of $x$ and $y$ to compare slopes. How can you tell if two lines are parallel just by looking at their equations?

Try simplifying $9x - 12y = 36$ first (for example, divide by 3). Then check each choice: which ones keep the same ratio between the $x$ and $y$ coefficients as the simplified equation?

For two equations to represent the same line, all three numbers (coefficients of $x$, $y$, and the constant) must be proportional. Which choice has matching $x$ and $y$ ratios but a different constant ratio?

In Desmos, type 9x - 12y = 36. This will graph the first line in the system.

Add each option as a new line in Desmos: -3x + 4y = 10, -3x + 4y = -12, 3x + 4y = 36, and 21x - 28y = 84. You can hide/show them using the checkboxes to focus on one pair at a time with the original line.

For each candidate equation together with 9x - 12y = 36, observe whether the lines intersect. If they cross at a point, the system has a solution. If they lie exactly on top of each other, there are infinitely many solutions. You are looking for the option where the line has the same slope as the original but never intersects it on the graph.

If you want to double-check, click on each line and have Desmos show them in slope-intercept form $y = mx + b$. The correct choice will have the same slope $m$ as the original line but a different $b$ (vertical intercept), confirming they are parallel and distinct.

For two linear equations in $x$ and $y$, the solutions are the points where their graphs (lines) intersect.

• One solution: lines intersect at exactly one point (different slopes).
• Infinitely many solutions: lines are the same line (all coefficients are proportional, including the constant).
• No solution: lines are parallel but different (the $x$ and $y$ coefficients are proportional, but the constants are not proportional).

The first equation is

Divide everything by $3$ to make the numbers smaller:

This has the same line as the original, just a simpler version. Now the pattern of coefficients is clear: $3$ with $x$, $-4$ with $y$, and constant $12$. Any parallel line must keep the same ratio $3 : -4$ for $x$ and $y$.

For a second equation to create no solution with $3x - 4y = 12$:

• Its $x$ and $y$ coefficients must be in the same ratio $3 : -4$ (so the slopes match and the lines are parallel).
• Its constant term must not match that ratio (so it is a different, parallel line, not the same line).

So we are looking for an equation that is a constant multiple of $3x - 4y$ on the left side, but not the same multiple on the right side.

First rewrite the given equation in the simpler form we found:

• Given: $9x - 12y = 36$ becomes $3x - 4y = 12$.

Now compare each option to $3x - 4y = 12$:

• A) $-3x + 4y = 10$ can be rewritten by multiplying both sides by $-1$:

The $x$ and $y$ coefficients match $3$ and $-4$ (same ratio), but the constant $-10$ is not $12$. So this is a parallel but different line → no solution.

• B) $-3x + 4y = -12$ becomes $3x - 4y = 12$ after multiplying by $-1$, which is exactly the same equation as the original → infinitely many solutions, not none.
• C) $3x + 4y = 36$ has coefficients $3$ and $4$, not $3$ and $-4$, so the slopes are different → one solution.
• D) $21x - 28y = 84$ divides by $7$ to give $3x - 4y = 12$, again exactly the same line → infinitely many solutions.

Only choice A) $-3x + 4y = 10$ gives coefficients in the same ratio for $x$ and $y$ but a different constant, so it is the correct second equation for a system with no solution.