When you already know the slope — one unknown, one formula.
The problem
A line with a slope of 3 passes through the points (6, 7) and (x, 1). Find the value of x.
Slope = (1 − 7) ÷ (x − 6) = 3, so −6 = 3(x − 6), giving 3x = 12 and x = 4. Check it: from (4, 1) to (6, 7) is right 2 and up 6 — slope 3. ✓ D moves the wrong way along the line.
The formula uses subscripts, so decide now which point is which.
Point 1
(6, 7)
gives
x1 = 6 y1 = 7
Point 2
(x, 1)
gives
x2 = x y2 = 1
x2 is the one we are hunting for.
You may call either point number 1. What you cannot do is take the top from one order and the bottom from the other — that flips the sign of the slope.
m = y2 − y1 x2 − x1
Rise over run — y on top, x on the bottom.
3 = 1 − 7 x − 6
1 − 7
x − 6
The slope 3 is already known, so it goes on the left. Everything else is just copying numbers into their slots.
3 = −6 x − 6
−6 divided by something equals 3.
A negative top with a positive slope means the bottom must be negative too. Expect x to be less than 6.
The unknown is stuck under a fraction bar. Get it out first.
(x − 6) on top cancels (x − 6) on the bottom, leaving just −6. The left side keeps the bracket because 3 was multiplied by it.
3(x − 6) = −6
from Step 4
x − 6 = −2
÷ 3 both sides
x = 4
+ 6 both sides
Dividing by 3 first is quicker than expanding to 3x − 18 = −6. Both routes give x = 4, but this one has fewer numbers.
A fraction slope. The line through (−2, 5) and (6, y) has slope −54. Find y.
(y − 5) ÷ (6 − (−2)) = −54
the run is 8
(y − 5) ÷ 8 = −54
clean up the bottom first
y − 5 = −10
× 8 both sides: 8 × (−54) = −10
y = −5
check: −10 ÷ 8 = −54 ✓
What is different: multiply both sides by the run (8), not by the 4 under the slope. The 4 then cancels by itself.
A horizontal line: slope 0. The line through (4, 2k − 1) and (−3, 15) is horizontal. Find k.
slope 0 → the rise is 0
a level line never climbs
2k − 1 = 15
the two y-values match
k = 8
+ 1, then ÷ 2
What is different: with slope 0 there is no fraction to clear — just set the two y's equal.
A vertical line: no slope. The line through (3m + 1, 2) and (−8, −5) is vertical. Find m.
vertical → the run is 0
the slope is undefined
3m + 1 = −8
the two x-values match
m = −3
− 1, then ÷ 3
What is different: an undefined slope has no number to put on the left of rise ÷ run, so the condition is "same x" instead.
The same letter in both points. The line through (3, k) and (k, 11) has slope 3. Find k.
(11 − k) ÷ (k − 3) = 3
the letter sits on top and on the bottom
11 − k = 3(k − 3) = 3k − 9
× (k − 3) both sides, then expand
20 = 4k, so k = 5
gather the k's on one side
(3, 5) to (5, 11): 6 ÷ 2 = 3 ✓
put k back into both points
What is different: dividing first does not help here, because k is on both sides. Clear the fraction, expand, and collect the k's. A letter in both x-values, like (2a, 5) and (a, 1), works the same way: the run is a − 2a = −a.
Horizontal means slope 0: the y's match. Vertical means no slope: the x's match. Setting the y's equal for the vertical line gives 2 = −5, which is impossible — a sign we picked the wrong condition.
Harder questions hide the slope inside a condition. Get the slope from the condition first, then Steps 2–5 run exactly as before.
| The question says | The slope to use |
|---|---|
| the three points are collinear (on one line) | the slope of the two points you know |
| parallel to another line | the same slope as that line (for y = mx + b, the slope is m) |
| perpendicular to a line of slope m | −1m: flip it and change its sign |
| the other line is given as ax + by = c | its slope is −ab (solve for y to see it) |
Worked: (2, −1), (5, 5) and (k, 13) are collinear. Find k.
m = (5 − (−1)) ÷ (5 − 2) = 6 ÷ 3 = 2
slope from the two known points
(13 − 5) ÷ (k − 5) = 2
the third point must give the same slope
8 = 2(k − 5), k − 5 = 4
multiply out the fraction, ÷ 2
k = 9
check: (5, 5) to (9, 13) rises 8 over 4 ✓
Worked: the line through (0, 1) and (4, y) is perpendicular to a line of slope 2. Find y.
m = −12
2 flipped is 12, then change the sign
(y − 1) ÷ 4 = −12
run from 0 to 4 is 4
y − 1 = −2, y = −1
check: slope −24 = −12, and 2 × (−12) = −1 ✓
Worked: a line of slope −3 passes through (1, 3). Find the area of the triangle it forms with the axes.
y-intercept (0, b): (b − 3) ÷ (0 − 1) = −3
on the y-axis, x is 0
b − 3 = 3, b = 6
× (−1) both sides
x-intercept (a, 0): (0 − 3) ÷ (a − 1) = −3
on the x-axis, y is 0
a − 1 = 1, a = 2
÷ (−3) both sides
area = ½ × 2 × 6 = 6
the two intercepts are the legs of a right triangle
For slope 2 the perpendicular slope is −12. Changing only the sign (−2) or only flipping (12) are the two answers set to catch you. Test: the two slopes must multiply to −1.
The plain "slope is 3, find x" version is rare; the exam wraps it in "collinear", "parallel to" or "perpendicular to", sometimes with the unknown in both points. Turn the words into a slope, set rise ÷ run equal to it, and check by recomputing the slope from your finished points.
slope = 62 = 3 ✓
Find y
A line with a slope of −2 passes through the points (3, y) and (7, 5). Find the value of y.
(5 − y) ÷ 4 = −2 gives 5 − y = −8, so y = 13. A drops the negative and solves 5 − y = 8. Sanity check: a negative slope falls as x grows, so the point at x = 3 must sit above 5.
Hint: this time the missing number sits on top of the fraction, and the slope is negative. Keep both minus signs.
