Point-Slope and Standard Form

In this lesson you will: Write the equation of a line from a point and a slope; Write the equation of a line through two given points; Move between point-slope, slope-intercept and standard form, and use intercepts.

Point-slope is the slope formula with the fraction cleared

Suppose a line has slope m and passes through the known point (x_1,y_1). Let (x,y) be any other point on it. The slope between the two is (y-y_1)/(x-x_1)=m.

Multiply both sides by x-x_1 and you have point-slope form. Nothing new is being claimed — it is the definition of slope, rearranged so that it is usable.

Two points: find the slope, then pick either one

Given two points, compute the slope first. Then feed either point into point-slope form. The two choices look different for one line of working and then agree.

For (-2,5) and (4,2) the slope is -1/2. Using (-2,5): y-5=-1/2(x+2), which simplifies to y=-1/2x+4. Using (4,2): y-2=-1/2(x-4), which simplifies to y=-1/2x+4 as well. Agreement is a free check.

Standard form hands you both intercepts

Standard form is Ax+By=C with A, B and C integers. Its advantage shows up at the axes: set x=0 and the x-term disappears, leaving the y-intercept; set y=0 and the same thing happens on the other side.

A.CED.2a asks you to create equations in two variables and graph them with labels and scales. Standard form is usually the natural way to write a situation where two quantities add to a fixed total.

Parallel and perpendicular, briefly

Two different lines with the same slope are parallel: same rate of change, so the gap between them never closes.

Two lines are perpendicular when their slopes multiply to -1 — each slope is the negative reciprocal of the other. Flip the fraction and change the sign: 2/3 pairs with -3/2, and 4 pairs with -1/4.

Common mistake — Losing the double negative in point-slope

For the line through (-3,4) with slope 2, a student writes y-4=2(x-3).

The eye copies the 3 and keeps the minus sign that is already printed in the formula. It looks like the formula, which is exactly why it survives a second glance.

The formula subtracts x_1, and here x_1=-3, so the parentheses is x-(-3)=x+3. The correct equation is y-4=2(x+3), or y=2x+10.

The student's version simplifies to y=2x-2, which at x=-3 gives -8 rather than 4.

How to catch it: Put the given point back into your finished equation. If (-3,4) does not satisfy it, the line you wrote is not the line you were asked for.