The form writes a line straight from one point and its gradient
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.
Make the line pass through (2, 3)
passing through (2, 3) forces 2m + c = 3, so each intercept c fixes m = (3 − c) / 2
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.
Graphs of equations · A Taxi Fare Read from a Straight-Line Graph
QuestionThe 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
Watch out for
- 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.
Your turn
1.Which point is y − 6 = 3(x − 4) written from?