Systems of linear equations: graph both, tap the intersection

An original SAT-style practice question on systems of equations, worked step by step in Desmos — the same graphing calculator that's built into Bluebook — with the algebra method alongside.

Practice question

3x + 2y = 7
5x − 4y = 41

If (x, y) is the solution to the system of equations above, what is the value of x + y?

⚡ Desmos is the fastest way here Both lines go in exactly as written — no rearranging into y = mx + b, no multiplying through, no sign slips.

Systems of equationsLinearMedium⏱ ~30 sec

Step by step

Type both equations as given

Desmos graphs equations in standard form directly, so there's nothing to solve for first. Put each equation on its own line.

3x+2y=7Blue line, sloping down

5x-4y=41Red line, sloping up

What you see: two straight lines crossing once, below the x-axis and to the right of the y-axis.

Tap the intersection

Tap where the lines cross. Desmos drops a grey point there and labels its coordinates — that point is the solution to the system.

(5,−4)So x = 5 and y = −4

What you see: a labelled point at (5, −4). Nothing else on the graph matters for this question.

Answer the question that was asked

The question wants x + y, not the point. Type it on a new line so the arithmetic is done for you.

5+(-4)= 1 — done

What you see: the row shows = 1 on the right. That's your answer.

Answer

The lines meet at (5, −4), so x + y = 5 + (−4) = 1. Check: 3(5) + 2(−4) = 7 ✓ and 5(5) − 4(−4) = 41 ✓.

⚠ Common trap

Stopping at the point and reporting 5 (just x) or −4 (just y). Another one: adding the two equations as they stand gives 8x − 2y = 48, which looks like progress but doesn't give x + y. Read the last line of the question again before you answer.

The algebra way, if you prefer

Elimination: multiply the first equation by 2 to get 6x + 4y = 14. Add it to 5x − 4y = 41 and the y terms cancel: 11x = 55, so x = 5. Substitute back: 3(5) + 2y = 7, so 2y = −8 and y = −4. That makes x + y = 1.

Elimination is quick here because the y coefficients (2 and −4) line up after one multiplication. When the numbers are messier — fractions, decimals, no easy multiple — graphing usually wins.

More walkthroughs: sum of a quadratic's solutions · vertex and maximum value · quadratic roots · circle equations · exponential growth · line of best fit · solving for x by graphing · linear inequalities · absolute value equations · Desmos cheat sheet · all walkthroughs

Stuck on a different practice question?

Snap a photo of any SAT-style math practice question and CalcHack walks you through the fastest route, in Desmos or by hand. Five free scans every day, no account needed to start.

Try it on your own question