How to solve systems of equations with Desmos on the SAT
Updated
Systems of equations show up on almost every digital SAT, and the built-in Desmos calculator can solve nearly all of them in under a minute. You don't rearrange anything or eliminate a variable. You type both equations, and the answer is the point where the graphs cross.
This guide walks through the method, four SAT-style examples (including the "no solution" and "how many solutions" kinds), and the mistakes that cost students points.
The method in three steps
- 1Type each equation on its own line, exactly as the question writes it. Desmos graphs as it is, so there's no need to solve for first.
- 2Click either graph. Desmos marks every point where the graphs cross with a gray dot.
- 3Click the gray dot to see its coordinates. The -coordinate is the solution for , and the -coordinate is the solution for .
That's the whole move. Every example below is a variation on it: the question just asks for something different about the crossing point.
Example 1: a linear system
Find x
Type on the first line and on the second. Click either line, then click the gray point where they cross.
Desmos shows , so . A quick check: and .
Answer:
Example 2: how many solutions?
Count the crossings
- A. Zero
- B. Exactly one
- C. Exactly two
- D. Infinitely many
Graph both equations and count the gray points. The line just touches the right side of the parabola, at one point.
One point of contact means one solution. In algebra terms, setting the sides equal gives , which is : a single root.
Answer: Exactly one
Example 3: no solution, with an unknown constant
Find the k that makes the lines parallel
Type both equations. Desmos will offer to add a slider for ; accept it. Drag the slider and watch the second line turn. The system has no solution when the lines are parallel, so they never cross.
At the lines are parallel and no gray point appears. To confirm with algebra: the slopes are and , and those are equal when .
Answer:
Example 4: a word problem
When do two costs match?
Write each total as a line, with as months: and . Graph both. At first you may see nothing useful, because the crossing is far from the origin. Zoom out, or set the window to from 0 to 10 and from 0 to 300.
They cross at : after 6 months, both gyms have charged $190.
Answer: months
Mistakes that cost points
- Answering with the wrong coordinate. The point is . If the question asks for , don't stop at .
- Not zooming out. Word problems and exponential equations often cross far off screen. If you see no crossing, zoom out before deciding there's no solution.
- Trusting a rounded decimal. Desmos may show for . On multiple choice, pick the choice that matches. On a typed answer, enter the fraction when you can.
- Using Desmos when the algebra is one step. If one equation already says , substitute. Graphing takes longer than that.
Practice this move
Practice on real SAT-style problems
Intersect Prep puts the same Desmos calculator beside 109 SAT-style problems that use this move, with a step-by-step strategy for each one. Create a free account to try a full lesson with the calculator; a pass unlocks the rest.
Questions
- Can you use Desmos for systems of equations on the SAT?
- Yes. The digital SAT has the Desmos graphing calculator built into every math question in the Bluebook app. Graphing both equations and clicking their intersection is allowed and is often the fastest method.
- Do I have to solve for y before typing an equation into Desmos?
- No. Desmos graphs equations in any form, such as 4x + y = 14, exactly as written.
- What does it mean if the graphs never cross?
- The system has no solution. For two lines, that means they are parallel: same slope, different intercepts.
- What if the two graphs are the same line?
- Then every point on the line is a solution, so the system has infinitely many solutions. In Desmos, one graph is drawn exactly on top of the other.