Every point on a line satisfies its equation. The point sitting on both lines satisfies both — the crossing IS the solution.
Read the graph
What is the solution of the graphed system?
The one point on both lines is the crossing, (2, 3) — x first, then y. A swaps the coordinates, the most common slip when reading a point. Check it: 2 + 1 = 3 ✓ and −2 + 5 = 3 ✓ — both equations are happy.
A line's graph is the set of every (x, y) pair that makes its equation true. Solving a system asks for the pair that makes both equations true at once — and the only point lying on both graphs is where they cross. Nothing else on either line qualifies.
crossing: (2, 3)
drop down for x, across for y
x = 2, y = 3
the system's one solution
Order matters: (2, 3) and (3, 2) are different points, and exams offer the swapped pair almost every time.
y = x + 1: 2 + 1 = 3 ✓
the gold line agrees
y = −x + 5: −2 + 5 = 3 ✓
so does the blue one
A point that only satisfies one equation lies on one line but off the other — it solves nothing. Both checks passing is what "solution of the system" means.
Which system is graphed, and using the crossing. One line crosses the y-axis at −1 and rises 1 for every 1 across; the other crosses at 8 and falls 2 for every 1 across.
y = x − 1 and y = −2x + 8
b from the y-axis crossing, m = rise ÷ run for each line
crossing (3, 2): 3 − 1 = 2 ✓, −6 + 8 = 2 ✓
the crossing must fit both equations you chose
y = −x + k through (3, 2): 2 = −3 + k → k = 5
a third line through the crossing: put the point in
line through (0, 0) and (3, 2): slope = 2 ÷ 3
from the origin, rise ÷ run is just y ÷ x
x-intercepts 1 and 4: area = ½ × 3 × 2 = 3
triangle with the x-axis: base between the x-intercepts, height = the crossing's y
exactly 1 solution
different slopes always cross once
0 solutions
same slope, different intercepts — never meet
infinitely many
every point works — one equation in disguise
These three are the only cases for two lines. Once a curve is involved, two crossings become possible — Step 6.
The crossing is not on a grid point. The graphs of y = x + 2 and y = −2x + 3 cross a little right of the y-axis.
x + 2 = −2x + 3
both equal y at the crossing
3x = 1 → x = 13
not a whole number
y = 13 + 2 = 73
from the first line
(13, 73)
check: −2(13) + 3 = 73 ✓
What is different: the picture only gives an estimate (about (0.3, 2.3)). Use it to pick the right region, then solve for the exact point.
Two equations, one line. How many solutions do y = 2x − 3 and 4x − 2y = 6 have?
−2y = −4x + 6
move the 4x across
y = 2x − 3
÷ (−2): the first equation again
infinitely many
every point of the line works
What is different: the equations look different, but one is the other rearranged and doubled, so the graphs lie on top of each other.
Different-looking equations are not different lines. Seeing two equations and answering "one solution" (or "none") without putting both into y = mx + b is the slip.
One line is vertical. Solve x = −2 and y = 3x + 4.
x = −2 already
the vertical line fixes x
y = 3(−2) + 4 = −2
substitute into the other line
(−2, −2)
check: on x = −2 ✓, and 3(−2) + 4 = −2 ✓
What is different: a vertical line has no slope, so the slope-comparison test does not apply — but it always crosses any line that is not vertical, exactly once.
The graph shows y = x² − 2x and y = x + 4. Find all the solutions.
x² − 2x = x + 4
both equal y at a crossing
x² − 3x − 4 = 0 → (x − 4)(x + 1) = 0
a quadratic: up to two answers
x = 4 or x = −1
matches the two crossings on the graph
(4, 8) and (−1, 3)
y from the line; check: 16 − 8 = 8, 1 + 2 = 3 ✓
distance between them = √5² + 5² = √50 = 5√2
if asked how far apart the solutions are: the distance formula
| Horizontal line y = k with y = x² − 2x (vertex (1, −1)) | Solutions |
|---|---|
| k > −1 (above the vertex) | 2 |
| k = −1 (just touches the vertex) | 1 — a tangent |
| k < −1 (below the vertex) | 0 |
The vertex and the intercepts of the parabola are clearly marked, so they turn up among the choices. A solution must sit on both graphs — test it in both equations. The same holds for a line and a circle: substitute the line into the circle and expect up to two points.
Expect the system read off a graph (which equations?), a line meeting a parabola or circle twice, a line y = k that just touches a vertex, or a follow-up that uses the crossing (a third line through it, a triangle with the axes). Read the crossing, then confirm it in both equations — ten seconds.
Read the graph
How many solutions does the graphed system have?
Both lines climb at the same slope, 1 — they are parallel and never meet, so the system has no solution. Algebra says the same: x + 1 = x − 2 collapses to 1 = −2, which is false for every x. C would need the two equations to be the SAME line.
