These Desmos tips for SAT systems of equations questions build on Parts 1 and 2 of this series, where we covered how to solve single-variable equations and equations with unknown constants, including the slider feature. In this part we focus on a new problem type: systems of equations, which appear frequently on the SAT Math section and can be solved in seconds with the right approach.
Systems of equations questions on the SAT fall into two categories. The first is linear systems, where both equations are straight lines and the solution is where they intersect. The second is mixed systems, where one equation is linear and one is quadratic. Both types are covered below with worked examples.
Desmos Tips for SAT: Solving Linear Systems of Equations
A linear system contains two straight-line equations. The solution is the point where the two lines intersect. Desmos finds this point visually and displays the exact coordinates when you hover over the intersection.
Example 1
Steps to solve a linear system using Desmos
Step 1.
Enter both equations into Rows 1 and 2 of the Desmos expression list.
Step 2.
Hover over the intersection point with your cursor. Desmos highlights the exact coordinates of the solution. 
The lines intersect at (15, -45). The question asks for the x-value of the solution, which is 15. Choice C is correct.
See Desmos in Action: Solving System of Equation
Check out this video where we demonstrate the step-by-step process of solving system of equations using the Desmos SAT calculator.
Desmos Tips for SAT: Solving Mixed Systems of Equations
A mixed system contains one linear and one quadratic equation. These are more complex because the graphs can intersect at zero, one, or two points depending on the values of any unknown constants. Desmos handles these with the slider feature covered in Part 2.
Example 2
Steps to solve a mixed system with an unknown constant
Step 1.
Enter both equations into Rows 1 and 2 of the Desmos expression list. When Desmos detects the unknown constant, it prompts you to add a slider. Click to add it. 
Step 2.
The question states the system has one solution, meaning the graphs touch at exactly one point. Use the slider to adjust the value of a until the two graphs intersect at exactly one point. The value a = -8 produces one intersection.
Step 3.
The value a = -8 tells you the constant, but read the question carefully. It asks for the x-value of the solution to the system. Hover over the intersection point to find x = 6. 
Desmos gets you to the answer faster than working through the algebra, but you still need to understand what the question is asking. The slider tells you the constant. The intersection coordinates tell you the solution. These are two different values and the question specifies which one it wants.
Additional Example Problems with Desmos SAT Calculator Solutions
Question 1
This is a mixed system where the linear function is a horizontal line with slope zero and a y-intercept of -1.5.
Solution
Using the Desmos slider, the value a = -14.5 produces exactly one solution. When a = -14.5, the graphs of the two functions intersect at only one point.
Question 2
This is a mixed system containing a vertical line at x = 8 and a quadratic equation symmetrical about the y-axis with a y-intercept of 8. There are no unknown constants, so no slider is needed.
Solution
Enter both equations into Desmos and hover over the intersection point. The graphs intersect at (8, 72). The question asks for the y-value, which is 72. Choice D is correct.
Question 3
This mixed system contains a quadratic equation with two unknown constants. Because the question is multiple choice, plug in each answer choice using sliders for constants a and b.
Solution
When a = -2 and b = 4, the graphs intersect at two points. This indicates the system has two solutions. Choice B is correct.
Key Takeaways for Using Desmos on Systems Questions
- For any system where both equations are known, enter them and hover over the intersection point to find the solution coordinates
- For systems with unknown constants, use sliders and adjust until the graph matches the condition stated in the question (one solution, two solutions, no solution)
- Always re-read whether the question asks for x, y, or the value of the constant. These are three different things Desmos can show you and students frequently pick the wrong one
- Mixed systems with two unknowns can be solved on multiple choice questions by plugging in each answer choice using two sliders
Frequently Asked Questions
Desmos can solve any system where you need to find intersection coordinates, identify the number of solutions, or determine the value of an unknown constant. The only cases where algebra may be faster are conceptual questions about whether a system has a solution in general, without specific numbers to work with.
Enter both equations into the Desmos expression list, then hover your cursor over the point where the two graphs cross. Desmos displays the exact coordinates of the intersection point. On the Bluebook version, tap the intersection point directly if using a touchscreen device.
Sliders represent unknown constants in an equation. When a question states that a system has exactly one solution, you adjust the slider until the two graphs touch at exactly one point. The slider value at that point is the value of the constant. This technique is covered in detail in Part 2 of this series.
A mixed system contains one linear equation and one quadratic equation. Unlike a linear system where two straight lines can only intersect at one point or not at all, a mixed system can have zero, one, or two solutions depending on where the line crosses the parabola. Desmos shows this visually, which is much faster than solving algebraically.
Systems of equations questions appear regularly in the Algebra and Advanced Math domains of the SAT Math section. Algebra accounts for approximately 35 percent of SAT Math questions, and systems are among the most frequently tested topics within that domain. See the full breakdown in the SAT test prep course guide.
Practice Desmos Systems Questions on Real SAT-style Problems
MentoMind’s SAT course includes 3,500 plus questions across Reading and Writing and Math, organized by topic and difficulty, with the built-in Desmos calculator available throughout just like the real exam.
