A point and a gradient fix a line
Through the point (2, 3) you can draw lines of every steepness. Ask for a gradient of 2 as well, and only one of them is left.
From (2, 3), a gradient of 2 means 1 across and 2 up, to (3, 5), then 1 across and 2 up again, to (4, 7). Going the other way, 1 back and 2 down, gives (1, 1). So the point and the gradient together fix every point of the line.
The line through the marked point (2, 3) with gradient 2: from (2, 3), every 1 across goes with 2 up.
Where the form comes from
Take any other point on the line and call it (x, y). From (2, 3) to (x, y), the run is x − 2 and the rise is y − 3.
The gradient is the rise divided by the run, and on this line the gradient is 2 everywhere. So . Multiply both sides by x − 2: y − 3 = 2(x − 2).
Every point on the line satisfies this equation, and no point off it does. It holds at (2, 3) itself too, where both sides are 0. Check with (4, 7): the left side is 7 − 3 = 4, and the right side is 2 × (4 − 2) = 4.
The lower marked point is (2, 3), and the upper one stands for any other point (x, y) on the line. The run between them is x − 2 and the rise is y − 3, and the rise divided by the run is always 2.
Point-slope form
The same steps work for any point and any gradient. The line through the known point with gradient m is . The small 1s mark the coordinates of that one known point. This is called point-slope form, because slope is another name for the gradient.
The y-coordinate of the point is subtracted from y, and the x-coordinate is subtracted from x. The line through (5, 4) with gradient −3 is y − 4 = −3(x − 5): the point and the gradient go straight into the form, with no working.
Watch the signs when a coordinate is negative. Through (−1, 2) with gradient 3, the form is y − 2 = 3(x − (−1)), which is y − 2 = 3(x + 1).
Back to y = mx + c
Expand the bracket and tidy up. From y − 3 = 2(x − 2): expand to get y − 3 = 2x − 4, then add 3 to both sides: y = 2x − 1. It is the same line, written in two ways. Now the y-intercept can be read off: the line crosses the y-axis at −1.
The same steps for y − 4 = −3(x − 5): expand to get y − 4 = −3x + 15, because (−3) × (−5) = 15. Add 4 to both sides: y = −3x + 19. Check with the point: −3 × 5 + 19 = 4.
passing through (2, 3) forces 2m + c = 3, so each intercept c fixes m = (3 − c) / 2
Make the line pass through (2, 3)
Make the line pass through (2, 3). Many lines do, one for each gradient. Set the gradient, which the handle calls the slope, to 2: the only intercept that then works is c = −1, the line y = 2x − 1.
The usual mistakes
Putting the coordinates with the wrong letters. For the point (2, 3), the 2 goes with x and the 3 goes with y: y − 3 = 2(x − 2), not y − 2 = 2(x − 3).
Adding the coordinates. y + 3 = 2(x + 2) is the line through (−2, −3). The form subtracts the coordinates of the point.
Expanding only part of the bracket. 2(x − 2) is 2x − 4, not 2x − 2.
Taking c to be the y-coordinate of the point. In y − 3 = 2(x − 2), c is not 3: expanding gives y = 2x − 1, so c = −1.
Worked example: A Taxi Fare Read from a Straight-Line Graph
Question The graph of a taxi fare, y dollars, against the distance traveled, x km, is a straight line. It passes through (2, 7) and (6, 13). (a) Find the equation of the line, and say what its gradient and its y-intercept mean for a passenger. (b) A journey costs $19. How long is the journey?
1.Find the gradient from the two points. From (2, 7) to (6, 13) the line rises 13 − 7 = 6 while it runs 6 − 2 = 4, so m = 64 = 1.5.
From (2, 7) to (6, 13) the line rises 6 while it runs 4, so the gradient is m = 64 = 1.5. 2.Use the point-slope form with the point (2, 7): y − 7 = 1.5(x − 2). Expand the bracket: y − 7 = 1.5x − 3, so y = 1.5x + 4.
Use the point (2, 7) in the point-slope form: y − 7 = 1.5(x − 2), which simplifies to y = 1.5x + 4. 3.(a) The equation is y = 1.5x + 4. The gradient 1.5 means that each kilometer adds $1.50 to the fare. The y-intercept 4 means that every journey starts with a charge of $4. Check with the other point: 1.5 × 6 + 4 = 13.
(a) The gradient 1.5 is the charge for each kilometer, $1.50. The y-intercept 4 is the $4 charged at the start of every journey. 4.For a fare of $19, put y = 19 into the equation: 19 = 1.5x + 4. Subtract 4 from both sides: 1.5x = 15. Divide both sides by 1.5: x = 10.
Put y = 19 into the equation: 19 = 1.5x + 4, so 1.5x = 15 and x = 10. 5.(b) The journey is 10 km long. Check: 1.5 × 10 + 4 = 19, so the point (10, 19) is on the line.
(b) The journey is 10 km long. The point (10, 19) is on the line.
Answer: (a) y = 1.5x + 4: each kilometer costs $1.50 and every journey starts at $4; (b) 10 km
Common mistakes
- Dividing the run by the rise, which gives 46. The gradient is the change in y divided by the change in x, because it measures how many dollars are added for each kilometer.
- Dividing $19 by 1.5 to find the distance. That treats the whole fare as a charge for distance. The $4 starting charge must be subtracted first, which leaves $15 for the distance.