The equation makes a table
The equation y = 2x + 1 is a rule: double x, then add 1. Choose values of x and the rule gives each y. When x = 0, y = 2 × 0 + 1 = 1. When x = 1, y = 3; when x = 2, y = 5; and when x = 3, y = 7. Written in rows, those pairs make a table of values.
The table of values of y = 2x + 1 for x = 0, 1, 2 and 3.
What the table shows
Read down the y column: 1, 3, 5, 7. Each time x goes up by 1, y goes up by 2. That constant gap is the gradient, the 2 in y = 2x + 1. The row where x = 0 gives y = 1, which is the 1 in the equation, the y-intercept.
Measure the gap only between rows where x goes up by 1. If a table jumped from x = 1 to x = 3, y would go up from 3 to 7, a gap of 4. The gradient is the change in y divided by the change in x: 4 ÷ 2 = 2, as before.
The table makes the graph
Each row of the table is a point: (0, 1), (1, 3), (2, 5) and (3, 7). Plot them on a grid and they line up. The straight line through them is the graph of y = 2x + 1, and the table is four of its points, taken at whole-number values of x.
The line also passes between those points. Every point on it satisfies the equation, including points the table left out: at x = 1.5, y = 2 × 1.5 + 1 = 4, so (1.5, 4) is on the line too.
The four rows of the table, plotted as (0, 1), (1, 3), (2, 5) and (3, 7), all lie on the line y = 2x + 1.
The graph gives back the equation
Now read the line to rebuild the equation. It crosses the y-axis at 1, so c = 1. Move 1 across along the line and it climbs 2, so m = 2. Put them into y = mx + c: y = 2x + 1, the equation we started from.
So any one of the three forms rebuilds the other two. The equation gives the table by substituting values of x. The table gives the graph by plotting each row as a point. The graph gives the equation by reading the gradient m and the y-intercept c.
passing through (2, 3) forces 2m + c = 3, so each intercept c fixes m = (3 − c) / 2
Make the line pass through (2, 3)
The line opens as y = 0.5x + 1: it crosses the y-axis at 1 and climbs 0.5 for each 1 across. The handle on the y-axis moves c, and the other handle tilts the line to change m. Make the line pass through the ring at (2, 3), and read its equation.
Is a point on the line?
A point lies on a line when its coordinates make the equation true. Is (4, 9) on y = 2x + 1? Put x = 4 into the equation: 2 × 4 + 1 = 9, which is the point's y-coordinate, so yes. Is (9, 4)? 2 × 9 + 1 = 19, which is not 4, so no. The first coordinate is always x.
The usual mistakes
Swapping the gradient and the intercept. The number that multiplies x is the gradient. A line that crosses the y-axis at 1 and climbs 2 for every 1 across is y = 2x + 1, not y = x + 2.
Losing the sign of the intercept. If the row for x = 0 reads 1, the line crosses the y-axis at +1, so the equation ends + 1, not − 1.
Leaving out the intercept. y = 2x passes through the origin. A line that crosses the y-axis at 1 needs the + 1.
A line from two points
A problem may give only two points of a line, with x more than 1 apart. The gradient is still the change in y divided by the change in x. Then the point-slope form , with either point as , gives the equation.
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.