Desmos for the SAT · Technique 2 of 15

Graph a System of Two Linear Equations

Two lines, one meeting point. Find it without touching algebra.

SolvesSystems of equationsPoint of intersectionBreak-even points

Here's the thing about a system of two linear equations: it always means the same picture. Two lines, and you want the one spot where they cross.

So type each equation into Desmos exactly as it is written, click the point where they meet, and read the coordinates. No substitution, no elimination, no arithmetic to fumble.

Break-even word problems are the same picture in disguise. Model each competing cost as its own line, say y = 20 + 0.10x for one plan and y = 15 + 0.15x for the other, and the x-coordinate where they cross is the break-even point where both cost the same.

Nova

Systems of two lines are everywhere on the Math section, and they're the perfect thing to graph. Two lines only cross in one spot, so instead of solving for x by hand, you just find the dot where they meet and read it off. It feels almost like cheating.

The algebra way vs the Desmos way

This technique sidesteps substitution or elimination.

The algebra way

Set the right-hand sides equal: 4x - 7 = -x + 8.
Collect terms and solve: 5x = 15, so x = 3.
Back-substitute into either equation: y = 4(3) - 7 = 5.

The Desmos way

Type y = 4x - 7 and y = -x + 8 as two lines.
Click where they cross and read (3, 5).

The payoff: The algebra has three chances to drop a sign or misadd; Desmos is two lines and one click on the point.

See the move, live

That same example, already loaded in the real calculator, the one built into Bluebook on test day. Try the move yourself.

Desmos graphing calculator, livepre-loaded

Before you start: set up Desmos

Convert decimals to fractions
Desmos shows decimals like 1.333. SAT choices are usually fractions. Click the round fraction-converter button next to a value to turn 1.333 into 4/3.
Turn on Projector Mode for clearer graphs
Desmos draws thin lines by default, which makes a crossing or intercept hard to pin down. Open the wrench icon (Graph Settings, top right) and turn on Projector Mode for thicker lines and larger labels, so the points you click are easier to hit exactly.
Widen the axis bounds when the graph looks empty
Desmos opens on a small window from -10 to 10 on both axes. If a crossing, vertex, or data point sits outside that box, the graph can look empty even when an answer exists. Open the wrench icon and widen the x and y bounds until the point you need comes into view.

Step by step

Nova

Follow along one step at a time, hit Next as you go. I'll flag the spots where people slip.

Step 1 of 4

Enter the first equation exactly as given, e.g. y = 4x - 7.

Desmos showing the single line y = 4x minus 7 graphed on the coordinate plane.

Now try it on a real SAT question

A system of two linear equations is given: y = 3x - 5 and y = -2x + 15.

What is the x-coordinate of the solution to the system?

Pick an answer to get instant coaching from Nova.

Nova

Both equations already start with "y =", so this is a graph-it question. Type both lines, click the crossing point, and read the x off the label.

Your turn, blank calculatornothing loaded

Set it up yourself in the blank calculator, then read the answer straight off the graph. Muscle memory beats watching.

Watch it

Graph two lines and click where they cross to solve the system.

The Desmos trap: when to close the calculator

If the question asks for a constant that makes the system have no solution or infinitely many solutions, sliders are slow and imprecise for that. Set the slopes (and, for infinite solutions, the intercepts too) equal algebraically instead.
If one equation is already solved for a variable and substitution takes ten seconds, that is faster than switching screens to type into Desmos.
For a nonlinear system, do not stop at the first crossing. Scroll out or hit Zoom Fit to check for a hidden second intersection, since the SAT often asks for "the positive value of x" or "the other solution," and that point can sit outside the default window.

Nova

Gut-check first, though: graphing isn't always the fast lane here. If a question asks for the value that makes the system have no solution or infinitely many, or one equation is already solved and substitution takes ten seconds, do that instead. Save the graph for when the crossing point is what you actually need.

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Put this move into real practice

"Graph a System of Two Linear Equations" 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.