How to Solve SAT Linear Systems with Desmos

Solve the original system 2x + y = 11 and x − y = 1. Enter each equation on its own expression line:

2x + y = 11
x - y = 1

You do not have to rearrange these linear equations before graphing. Use the official testing calculator for practice and confirm the workflow in Bluebook.

Select a line and then the intersection point. For these equations the point is (4,3), so x = 4 and y = 3. Check both originals: 2(4) + 3 = 11 and 4 − 3 = 1. If the question asks for x + y, answer 7; if it asks for x, answer 4. A coordinate pair and a requested scalar are different answers.

Add 2x + y = 11 and x − y = 1. The y terms cancel, giving 3x = 12 and x = 4. Substitute into x − y = 1 to get y = 3. This confirms the graph and avoids dependence on rounded coordinate labels.

• Different slopes: exactly one intersection, hence one solution.
• Equal slopes and different intercepts: parallel lines, hence no solution.
• Equivalent equations: the same line, hence infinitely many solutions.

For example, y = 2x + 1 and y = 2x + 4 are parallel. The equations y = 2x + 1 and 2y = 4x + 2 represent the same line. Verify algebraically when two graphs appear to overlap.

Enter 3x + y = 17 and x − y = 3. Before revealing a point, add the equations to obtain 4x = 20. Thus x = 5 and y = 2, and x + y = 7. Check 3(5) + 2 = 17 and 5 − 2 = 3. If the intersection is outside your view, adjust the window.