Solve by Graphing

System of Equations with a Parameter on the SAT: Desmos Walkthrough

You'll learn to find the unknown value in a system of equations by putting it on a slider and graphing until the lines meet the required condition (one solution, no solution, or infinitely many).

This lesson solves by graphing: turn the equation into curves, then read intersections, vertices, or tangency straight off the Desmos graph.

Take this lesson interactively

A real SAT-style question

$a x + b y = 72$

$6 x + 2 b y = 56$

In the given system of equations, $a$ and $b$ are constants. The graphs of these equations in the xy-plane intersect at the point $left parenthesis 4 comma y right parenthesis$ . What is the value of $a$ ?

$3$

$4$

$6$

$14$

How to solve it in Desmos, step by step

The idea

The intersection sits at x = 4. Pick any value for b, graph the second equation to read the y-value there, then add a slider on a so the first line passes through that same point.

1
Step 1
Add a slider for b and set it to 1. Any value works because the crossing height at x equals 4 is fixed.
Type in Desmos
$b=1$
2
Step 2
Type the second equation. Desmos draws this line.
Type in Desmos
$6x+2by=56$
3
Step 3
Type the vertical line at x equals 4. Click the gray dot where it crosses the line you just drew. Desmos shows the point 4 comma 16, so the shared y-value is 16.
Type in Desmos
$x=4$
4
Step 4
Type the first equation. Desmos draws this line and adds a slider for a.
Type in Desmos
$ax+by=72$
5
Step 5
Drag the a slider until the first line passes through the gray dot at 4 comma 16. It lands exactly there when a reads 14, which is choice D.

The answer is D.

$14$

Now you try one

Same question type, different numbers. Use the exact Desmos moves from the walkthrough above.

a x + b y = 90

9 x + 3 b y = 60

In the given system of equations,

a

and

b

are constants. The graphs of these equations in the

xy

-plane intersect at the point

(5, y)

. What is the value of

a

Practice this with a live Desmos calculator

The interactive version of this lesson builds the Desmos graph up step by step as you click through, with a live calculator you can type into and your progress saved.

Open the interactive lesson