Every point on a line makes its equation true. The point on both lines makes both true — the crossing is the solution.
What is the solution to the system?
The graph of a system of two linear equations is shown in the xy-plane. What is the solution (x, y) to the system?
The crossing does not land on a corner of the grid, so the picture alone cannot finish this one. Double the second equation and add: 4x − 2y = 6 plus x + 2y = 6 gives 5x = 12, so x = 2.4 and then y = 6 − 2.42 = 1.8. Check both: 2.4 + 3.6 = 6 ✓ and 4.8 − 1.8 = 3 ✓. Every other choice sits on the gold line x + 2y = 6 and fails the blue one — (2, 2) is the nearest grid corner and gives 2x − y = 2, (2.5, 1.75) gives 3.25, (3, 1.5) gives 4.5. Landing on one line is not a solution of the system.
1
The crossing is the solution, and nothing else is.
A line's graph is every (x, y) that makes its equation true. A system asks for the pair that makes both true, so the answer has to be on both lines at once.
That is why a point sitting neatly on one line is worth nothing on its own. It satisfies one equation and fails the other, which is exactly the shape of the wrong answers in the opening item. Only the intersection qualifies, and on the SAT that point is what “the solution to the system” always means.
on both lines → solves both equationson one line → solves oneon neither → solves nothing
2
Read the crossing off the grid — x first, then y.
Drop straight down from the point to find x, then run straight across to find y. Here the lines meet on a corner of the grid, so the reading is exact.
straight down from the crossing → x = −3
three gridlines left of the y-axis
straight across from the crossing → y = 2
two gridlines above the x-axis
solution: (−3, 2)
always written x first
The swapped pair is on the answer list
(−3, 2) and (2, −3) are different points, and the SAT offers the swap on almost every item of this shape. Say the coordinates out loud as “x is −3, y is 2” before you look at the choices, and watch the signs: left of the y-axis is negative x, below the x-axis is negative y.
3
Run it backwards: turn the picture into the system.
Some items hand you the graph and ask which system it shows. Read each line separately — where it cuts the y-axis, then how it steps.
Gold line
Blue line
cuts the y-axis at
5
−1
from there to (−3, 2)
left 3, down 3
left 3, up 3
slope
−3−3 = 1
3−3 = −1
y = mx + b
y = x + 5
y = −x − 1
standard form
x − y = −5
x + y = −1
Moving a term flips its sign
y = x + 5 becomes x − y = −5, not x − y = 5 and not x + y = 5. Subtract y from both sides first, then subtract 5: you are left with x − y = −5. When the choices are four systems in standard form, this single sign is usually the only thing separating two of them.
4
Check the point in BOTH equations before you move on.
One substitution proves nothing. Two passing substitutions is what “solution of the system” means.
x − y = −5: (−3) − 2 = −5 ✓
the gold line agrees
x + y = −1: (−3) + 2 = −1 ✓
so does the blue one
And the algebra lands on the same point, which is the real reason the picture is trustworthy.
add the equations2x = −6the y terms cancel
solvex = −3then −3 + y = −1 → y = 2
(−3, 2) — read from the grid, confirmed by elimination
5
Count the solutions on sight.
Before you calculate anything, look at the picture. There are only three things two straight lines can do.
crossing → exactly oneparallel → noneone line drawn twice → infinitely many
The third picture is the one people miss, because there is nothing on the screen to tell you two equations were graphed rather than one. If a stem says a system has infinitely many solutions, it is telling you the two equations describe the same line — every coefficient and the constant share one ratio.
6
When the crossing misses the grid corners, stop reading and solve.
That is the opening problem. The lines meet between gridlines, so “about 2 and a bit” is as far as the eye can take you — and the SAT puts the nearest grid corner on the answer list for exactly that reason.
x + 2y = 6 and 2x − y = 3
the two labelled lines in the figure
2 · (2x − y = 3) → 4x − 2y = 6
set up to cancel the y terms
(x + 2y) + (4x − 2y) = 6 + 6
add the two equations
5x = 12, so x = 2.4
a calculator is allowed — 12 ÷ 5
2.4 + 2y = 6 → 2y = 3.6 → y = 1.8
back-substitute into the simpler equation
check: 4.8 − 1.8 = 3 ✓
the equation you did not use
Desmos reads the crossing exactly
Type x + 2y = 6 on one line and 2x − y = 3 on the next — Desmos graphs standard form without rearranging — then click where the two curves meet and the point appears as (2.4, 1.8). It works just as well when a graph is printed in the stem: retype the two equations, or the two points you can read off a line, and let the calculator find the intersection instead of squinting at the grid. Zooming the printed figure never adds precision that was not drawn in.
✓
Check: picture, point, proof.
read itx first, then y(−3, 2), never (2, −3)
test itboth equations, not one−3 − 2 = −5 ✓ and −3 + 2 = −1 ✓
off the corners5x = 12 → x = 2.4solve or use Desmos ✓
Remember the flow
Crossing = solution · x then y · Check in both · Parallel means none
If the point is not on a grid corner, the graph is a hint, not an answer.
Now you try
Practice problem
What is the value of x + y?
The graph of a system of two linear equations is shown in the xy-plane. If (x, y) is the solution to the system, what is the value of x + y?
Double the first equation to match the y terms: 6x − 4y = 10, and add x + 4y = 4 to get 7x = 14, so x = 2. Then 2 + 4y = 4 gives y = 0.5, and x + y = 2.5. Check the other equation: 6 − 1 = 5 ✓. D is 3, the answer you get by reading the crossing as the nearest grid corner (2, 1) — the y-value is half a unit lower than that, which the grid cannot show. B is 2, the value of x on its own: the question asked for the sum, not the point. A is 1.5, which is x − y instead of x + y.
Hint: the crossing sits on the vertical gridline x = 2 but between two horizontal ones, so read x off the picture, then get y from an equation rather than from the grid. Then read the question one more time — it wants a single number, not an ordered pair.
