Desmos for the SAT · Technique 2 of 23
Solve a System of Equations by Graphing
Two graphs, and the solutions are wherever they meet. Which pairs can meet twice, once, or never.
Watch the lesson
The whole move worked on one real SAT question, narrated.
A system of equations always means the same picture: two graphs, and the solutions are wherever they meet. Two lines that are not parallel meet once. A line and a parabola can meet twice, once, or not at all, and Desmos shows you which: count the gray dots where the graphs cross and you have counted the solutions. Desmos also dots intercepts and the vertex, so count only the dots where the two graphs meet, then click the one the question asks about.
Most systems on the test arrive as a story rather than as two equations: two ticket prices and a total, two purchases of the same two goods, a trip split between highway and local roads, two phone plans with different rates. Give each unknown a letter and each sentence becomes an equation. That is the only hard step; the graphing is the same. For a break-even question, model each cost as its own line, say and , and the x-coordinate of the crossing is where both cost the same.
This is also the lesson that answers how many. Counting solutions is counting crossings, and how many crossings are possible depends only on which two shapes you were handed. The two tables below are the whole rule. Once you know a line and a circle can only meet twice, once, or not at all, a question claiming three solutions has answered itself.
Nova
Systems are everywhere on the Math section, and they are the perfect thing to graph. Two lines that are not parallel meet once; a line and a parabola can meet twice, once or never, and Desmos puts a dot on every crossing. Count the dots, click the one you need, read it off.
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 substitution into a quadratic, then factoring.
The question
and
A solution to the system is (x, y). What is the greatest possible value of x?
The algebra way
- Set the two right sides equal:
x + 9 = x^2 + 16x + 63. - Rearrange to
x^2 + 15x + 54 = 0and factor it:(x + 6)(x + 9) = 0. - Read off
x = -6orx = -9and pick the greater.
The Desmos way
- Type both equations as given and select one: two dots appear where they meet.
- Click the crossing further to the right and read
x = -6.
The payoff: The graph shows how many solutions there are before you touch any algebra, and both crossings are labeled, so picking the greater x is one click.
Quick reference
What you see on screen, and what it means
| What you see | Number of solutions |
|---|---|
| The graphs cross at two gray dots | Two |
| No dot where they meet, the graphs only touch (tangent) | Exactly one. Desmos marks crossings, not touch points, so a tangent shows as the two curves meeting with no dot there. Confirm it with the algebra. |
| The graphs never touch | None |
| The two graphs merge into one | Infinitely many |
How many times each pair can meet
| The pair | Possible number of meetings | What the extremes look like |
|---|---|---|
| Line and line | 0, 1, or infinitely many | Parallel and apart is none; the same line twice is infinitely many. Two distinct non-parallel lines always meet exactly once, so a pair of lines can never meet twice. |
| Line and parabola | 0, 1, or 2 | Clear of the curve is none; only touching the curve (tangent) is one; straight through it is two. |
| Line and circle | 0, 1, or 2 | Missing the circle is none; touching the edge (tangent) is one; cutting across it is two. Never more, however the line is angled. |
| Line and exponential | 0, 1, or 2 | An exponential curve can be missed, touched once, or cut twice. A horizontal line is the special case: it meets the curve once if it sits above the asymptote and never if it sits on or below it. |
Step by step
The whole move, all 5 steps
- Type the first equation on its own line, exactly as given.
- Type the second equation on the next line. It does not matter whether they are two lines or a line and a curve.
- Click inside either equation box to reveal the gray dots. Count the dots where the graphs meet: that is how many solutions the system has.
- Click each crossing to read its coordinates.
- Answer what the question asks for: x, y, the pair, the greatest x, or something built out of both.
The question
A system of two equations is given: and .
A solution to the system is (x, y). What is the greatest possible value of x?
- A-9
- B-6
- C3
- D6
Step 1 of 6
Type the first equation exactly as given.
In the calculator: y=x+9
Now try it on a real SAT question
Head or scratch paper. A story problem, but read it and the second quantity is already pinned down: two nerite snails is $2, so $10 of rabbit snails is left. That is mental arithmetic. Recognizing a system in a word problem matters; opening the calculator for this one does not.
Naomi bought both rabbit snails and nerite snails for a total of $12. Each rabbit snail costs $2 and each nerite snail costs $1.
If Naomi bought 2 nerite snails, how many rabbit snails did she buy?
Pick an answer to get instant coaching from Nova.
Give each kind of snail a letter and the sentences become two equations. One of them already tells you a quantity outright, so how much work is really left?
Two nerite snails at $1 each account for $2 of the $12, leaving $10 for rabbit snails at $2 each, so she bought 5. As a system: with meets at .
In the app, drills like this adapt to your level, and your AI coach can walk you through any step you get stuck on.
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
- 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.
If the calculator does something you did not expect
- For a nonlinear system, do not stop at the first crossing. Zoom out (scroll, or the minus button) 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.
Put this move into real practice
"Solve a System of Equations by Graphing" 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 freeSAT 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.