How do you find the equation of the line through two given points?

Two numbers and nothing more

A line is fixed entirely by two numbers: the slope m and the y-intercept n. Every question about a line comes down in the end to finding those two.

The slope says how much y rises when x increases by one. A slope of 3 means a rise of 3 for each step to the right; a slope of minus a half means a fall of a half.

n is the value of the function when x is zero, so it is read straight off the graph where the line crosses the vertical axis. In a word problem it is almost always the starting state: the fixed charge, the amount already in the account.

Reading it off the graph

When a sketch is given there is nothing to calculate. Read n at the y-intercept. Read the slope by picking two clear points on the line and counting the steps between them: how far up for how far across.

Pick points that land on grid intersections. Two points read approximately give a slope that is approximately right, which is exactly enough to lose the mark.

A falling line has a negative slope, a horizontal line has slope zero, and a vertical line is not a function at all and has no defined slope.

Parallel, crossing, and the meeting point

Two lines are parallel if and only if their slopes are equal. If n is equal too they are the same line. If the slopes differ the lines meet at exactly one point.

Find that point by setting the two expressions for y equal to each other. Solve for x, then substitute into either equation to get y.

In a word problem the crossing point is the moment when two options cost the same: after how many months the two phone plans are equal, beyond what distance the second taxi is cheaper. That is where the real question lives.

Worked examples

1. Find the equation of the line through (2, 5) and (6, 13).

   1. Slope: 13 minus 5 over 6 minus 2, which is 8 over 4
   2. So m equals 2
   3. Substitute (2, 5): 5 equals 2 times 2 plus n
   4. 5 equals 4 plus n, so n equals 1

   Answer: y equals 2x plus 1

2. Where do y equals 3x minus 4 and y equals minus x plus 8 cross?

   1. Set them equal: 3x minus 4 equals minus x plus 8
   2. Rearrange: 4x equals 12
   3. x equals 3, substitute into either line

   Answer: They cross at (3, 5)

3. Plan A costs 30 a month plus 2 per hour. Plan B costs 50 plus 1 per hour. From when is B cheaper?

   1. Plan A: y equals 2x plus 30
   2. Plan B: y equals x plus 50
   3. Set equal: 2x plus 30 equals x plus 50
   4. x equals 20

   Answer: Beyond 20 hours a month, plan B is cheaper

Common mistakes

Computing the slope with the order reversed in one place

If the numerator subtracts the first point from the second, the denominator must do the same. Reversing one of them flips the sign, giving a line that falls where it should rise.

Swapping the x-intercept and the y-intercept

For the y-intercept set x to zero; for the x-intercept set y to zero. Confusing them produces the right point for the wrong question.

Concluding lines are parallel because they look it

Parallelism is decided by equal slopes, not by appearance. Slopes of 2 and 2.1 look almost identical on a sketch and certainly do cross, just outside the part that was drawn.

More in Mathematics, three units

• How do you turn a word problem into an equation without going wrong at the start?
• When do you use a distance-speed-time table and when the part done in one hour?
• How do you sketch a parabola without computing a whole table of values?
• Why does a 10 percent rise followed by a 10 percent fall not return the original price?
• Two classes share an average — what else do you need to compare them?
• How do you find an average from a frequency table rather than from a list?