Systems with Unknown Constants

In the last video, you saw a system with an unknown constant — something like “for what value of k does this system have no solution?” — and we solved it algebraically by matching coefficients. That method works. But for most of these questions, Desmos is faster and harder to mess up. This article is about learning that approach: using sliders to see the answer instead of grinding through the algebra to find it.

These are some of the most common hard questions on the SAT, and they trip students up because they feel unfamiliar. Once you see how they work on Desmos, they stop being intimidating.

What no solution and infinitely many actually mean

This was covered in the video, but it's worth having on the page because everything in this article depends on it.

Every system of two linear equations does one of three things:

• One solution — the two lines cross at exactly one point. This is the normal case. Most systems you've solved so far have been this.
• No solution — the two lines are parallel. Same slope, different intercepts. They never touch, so there's no point that satisfies both equations. On Desmos, you'll see two lines running in the same direction that never meet.
• Infinitely many solutions — the two equations describe the same line. Same slope, same intercept — one equation is just a scaled version of the other. Every point on the line satisfies both equations. On Desmos, the two lines overlap completely and you'll only see what looks like one line.

Recognizing these questions

The giveaway is a letter in the system that isn't x or y. You'll see something like k, h, a, or t sitting in one of the coefficients or constants. And the question will ask something like:

• “For what value of h does the system have no solution?”
• “What value of k makes the system have infinitely many solutions?”
• “For what value of a does the system have exactly one solution?”

If you see an unknown constant plus language about the number of solutions, you're dealing with this question type.

Solving algebraically

The previous video walked through how to solve these by matching coefficients — setting up ratios and solving for the unknown constant. That approach works, and it's worth understanding conceptually. But for most of these problems on the SAT, Desmos is a better and faster approach. Here's how.

The Desmos slider approach

This is the payoff we've been building to since the Desmos article earlier in this unit.

Here's the process. Say you're given this system:

And the question asks: for what value of h does the system have no solution?

1. Type both equations into Desmos exactly as they appear. When you type the first equation, Desmos will see the letter h and ask if you want to “add a slider” for it. Click yes.
2. A slider bar appears at the top of your expression list, with h on a draggable dot and a range (usually −10 to 10 by default). The second equation — 4x + 3y = 15 — doesn't move; it's fixed. The first equation changes shape as you drag h, because h controls the slope of that line.
3. Drag the slider slowly and watch the first line tilt. You're looking for the moment the two lines become parallel — same angle, running in the same direction, but not touching. When that happens, the system has no solution.
4. Read the value of h off the slider. That's your answer.

For this example, when you drag h to 4, the first line locks into the same slope as the second. The two lines are parallel — same slope, different y-intercepts — so the system has no solution. h = 4.

For infinitely many solutions, the process is identical except you're looking for something different: drag until the two lines overlap completely. When they merge into what looks like a single line, every point satisfies both equations — that's infinitely many solutions.

Worked examples

When Desmos struggles

Everything above works beautifully when there's one unknown constant. Drag one slider, watch for parallel or overlap, read the value. That covers the vast majority of these questions on the SAT.

But occasionally you'll see a system with two or even three unknown constants — something like ax + by = 10 paired with another equation. Now you'd need to drag two sliders simultaneously and find the combination that makes the lines parallel or identical. For those problems, the algebraic approach from the video — matching coefficient ratios — is genuinely faster and more reliable.

Quick practice

Try these on your own. Pick an answer and you'll see right away whether it's correct, along with the reasoning.