Find the slope, feed in a point, then dress the answer in whichever form the choices are wearing.
Which equation defines line t?
Line t in the xy-plane passes through the points (−2, 9) and (4, −3).
m = −3 − 94 − (−2) = −126 = −2, then y − 9 = −2(x + 2) gives y = −2x − 4 + 9 = y = −2x + 5. Both points check: −2(−2) + 5 = 9 and −2(4) + 5 = −3 ✓ A writes y − 9 = −2(x − 2), using +2 where the point's x is −2, and lands on 13. C loses the sign of the slope: y − 9 = 2(x + 2) gives y = 2x + 13, a line that rises where this one falls. D divides run by rise: 6−12 = −12 is the reciprocal of the slope, not the slope.
m = y₂ − y₁x₂ − x₁
m = −3 − 94 − (−2)
(−2, 9) first on both top and bottom
= −126
4 + 2 = 6 underneath
m = −2
down 2 for every 1 across
Whichever point you call the first one, use it first on both the top and the bottom. Mixing them flips the sign and the line leans the wrong way. And rise goes on top: run over rise gives −12 here, which is choice D in the opening problem.
y − y₁ = m(x − x₁). Drop the numbers straight in — there is no b to solve for.
y − 9 = −2(x − (−2))
using (−2, 9); the form subtracts x₁
y − 9 = −2(x + 2)
a negative x₁ becomes a plus
y − 9 = −2x − 4
distribute
y = −2x + 5
+ 9 both sides
Using the other point gives y + 3 = −2(x − 4), which tidies to y = −2x + 8 − 3 = −2x + 5 — the same line. Either point is fine; pick the one with the friendlier signs.
The SAT sometimes leaves an answer in point-slope form, untidied. y − 9 = −2(x + 2) is a complete equation of the line, and if it is among the choices you are already finished.
y = mx + b: m is the steepness and the lean, b is where the line crosses the y-axis.
falls 2 for every 1 to the right
crosses the y-axis at (0, 5)
When the given point is already on the y-axis, there is nothing to solve: a line with slope −3 through (0, 7) is y = −3x + 7, read straight off.
slope 4, through (3, 1): 1 = 4(3) + b
substitute the point into y = mx + b
1 = 12 + b, so b = −11
solve for the intercept
y = 4x − 11
check: 4(3) − 11 = 1 ✓
For the x-intercept of this line, set y = 0: 4x = 11, so x = 114.
In Ax + By = C: the x-intercept is CA, the y-intercept is CB, and the slope is −AB. Cover a term with your thumb and divide.
3x + 4y = 24
standard form, A = 3, B = 4, C = 24
y = 0: 3x = 24, x = 8
x-intercept (8, 0), and 243 = 8
x = 0: 4y = 24, y = 6
y-intercept (0, 6), and 244 = 6
slope = −AB = −34
from (0, 6) to (8, 0): −68 = −34 ✓
This is why the SAT writes so many linear models in standard form. An item that asks for the sum of the intercepts, or for the value that makes the graph cross the x-axis at a given place, is one division away — there is no need to solve for y first.
| Form | Looks like | Reach for it when |
|---|---|---|
| point-slope | y − y₁ = m(x − x₁) | you have a point and a slope |
| slope-intercept | y = mx + b | you want the slope or the y-intercept |
| standard | Ax + By = C | the choices are written that way, or you want the intercepts |
y − 9 = −2(x + 2)
point-slope
y = −2x + 5
expand and tidy → slope-intercept
2x + y = 5
+ 2x both sides → standard
y = 5 − 2x1 = −2x + 5
and back again, to be sure
Fractions clear the same way. y = (34)x + 2 becomes 4y = 3x + 8, then 3x − 4y = −8. Multiply every term, including the one that was already whole.
“Which equation represents this situation?” often offers four differently-dressed equations, more than one of which looks plausible. Put both into the same form and compare.
6x − 4y = 10 and y = (32)x − 52
same line, or not?
−4y = −6x + 10
move 6x across
y = (32)x − 52
divide every term by −4
identical → the same line
same slope and same intercept
In standard form there is a faster test: two equations describe the same line exactly when one is a constant multiple of the other, all three numbers included.
| Compare with 6x − 4y = 10 | Ratios | Verdict |
|---|---|---|
| 9x − 6y = 15 | 96 = 64 = 1510 = 1.5 | the same line |
| 9x − 6y = 12 | 1.5, 1.5, but 1210 = 1.2 | parallel, a different line |
| 6x + 4y = 10 | signs disagree | a different slope entirely |
9x − 6y = 12 shares its slope with 6x − 4y = 10 and nothing else; its y-intercept is −2 while the original's is −52. If you are unsure, take one point: (1, −1) satisfies 6x − 4y = 10 because 6 + 4 = 10, but 9(1) − 6(−1) = 15 ≠ 12. One point settles it.
Desmos accepts standard form as typed. Enter both equations and look: one line means they are the same, two lines means they are not, and if they cross you can read the point. Checking four choices this way takes about fifteen seconds.
Testing only the point you substituted proves nothing — it was true before you started. Check the other one.
Which equation defines the same line in the xy-plane?
y − 3 = 25(x + 10)
Expand: y − 3 = (25)x + 4, so y = (25)x + 7. Multiply every term by 5: 5y = 2x + 35, and moving the x term across gives 2x − 5y = −35. Check the point the form hands you, (−10, 3): 2(−10) − 5(3) = −35 ✓ B has the sign of the constant wrong — it is the line through (10, −3) instead. C swaps the coefficients, which turns the slope 25 into 52. D changes the sign on the y term, giving slope −25, a line falling where this one rises.
Hint: get the given equation into y = mx + b first, then convert once. Or take the point (−10, 3) that point-slope form is showing you and test it in all four choices — only one survives.
