Desmos for the SAT · Technique 15 of 23

Use a Slider to Test an Unknown Constant

Drag a value around until the graph does what the question describes.

SolvesUnknown constantFind the value that fits

Watch the lesson

The whole move worked on one real SAT question, narrated.

0:00 / 0:00

The same question, start to finish: add the slider, drag until the line sits on (4, 2), then check the value on one more row.

Some questions give you an equation with an unknown constant, like a in ax + 4y = 20, and ask for the value that makes some condition true, like passing through a given point.

Type the equation in with a letter standing in for the unknown, and Desmos offers to turn that letter into a slider. Drag it until the graph matches the condition, and read the value off the slider.

Which pairs of shapes can meet twice, once or never is the reference table in the systems lesson. When the condition is about the number of solutions, read the intersections as you drag: one gray dot where the graphs touch is exactly one solution (tangent), graphs that never touch is none, and two graphs that merge into one is infinitely many.

Nova

When a question hands you an equation with a mystery letter in it, you don't have to solve for it, you just make it a slider and drag until the graph does exactly what the question asks. Watch the line swing into place as you slide.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps solving for the unknown constant algebraically.

The question

ax + 4y = 20, passing through the point $(4, 2)$

What is the value of a?

The algebra way

Substitute the point into the equation: a(4) + 4(2) = 20.
Simplify: 4a + 8 = 20, so 4a = 12.
Divide by 4: a = 3. Divide the 20 by 4 before the 8 comes across and you land on 5, which is one of the choices.

The Desmos way

Type ax + 4y = 20 and add a slider for the letter a.
Plot the point (4, 2) and drag a until the line passes through it, landing on a = 3.

The payoff: The slider shows what a is doing, and one substitution row confirms the exact value. The slider earns its keep on the conditions you cannot substitute your way to in one line: two lines parallel or identical, or a line tangent to a curve.

Step by step

The whole move, all 4 steps

Type the equation using a letter other than x or y for the unknown constant.
Click “add slider” when Desmos offers it. That letter is now a knob you can turn.
Plot the point, or set up the condition, that the question describes.
Drag the slider until the condition is satisfied, then read the value off the slider.

The question

The equation ax + 4y = 20 is graphed in the xy-plane, where a is a constant. The graph passes through the point $(4, 2)$ .

What is the value of a?

Step 1 of 4

Type the equation using a letter other than x or y for the unknown constant: ax + 4y = 20, with the a left in.

In the calculator: ax+4y=20

Syntax: Use a letter other than x or y, e.g. $ax + 4y = 20$ , then click "add slider: a" and drag a to test the condition.

Now try it on a real SAT question

Head or scratch paper. One substitution and it is over: $f(7)$ is $4\times 7$ plus b, which is $28+b$ , and the question says that equals 28. A slider would get you there too, but building it takes longer than the arithmetic does.

The linear function f is defined by $f(x)=4x+b$ , where b is a constant, and $f(7)=28$ .

What is the value of b?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

A slider can reveal two values that both fit, like $k = 4$ and $k = -4$ . Reread the question for a constraint like k > 0 to choose the right one.
If the condition is a solution count, confirm the exact value with the discriminant: $b^{2} - 4ac = 0$ means exactly one solution (tangency).
When the constant only has to satisfy one point, like this one, the slider shows the answer and one substitution row confirms it in about the same time. Save the drag for conditions that are a picture: two lines parallel or identical, or a line tangent to a curve.

If the calculator does something you did not expect

Sliders default to a range of -10 to 10, so a large constant like 25 or 50 sits past the end. Click the small numbers at the ends of the slider to widen the range.

Nova

A slider lands near a value, not on it. Drag until the picture is right, two parallel lines or one line, then take the exact number from the ratio of the coefficients. The drag finds the neighborhood; the ratio is the answer.

All Desmos techniques

Put this move into real practice

"Use a Slider to Test an Unknown Constant" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.