📏Forms of a Line

Every straight line can be written in more than one way, and the three standard ways are not rivals — each one puts a different fact where you can read it at a glance.

• ◆Slope-intercept form, y = mx + b. The slope m and the height b where the line cuts the y-axis are both sitting in plain sight. This is the form to graph from, and the form to compare two lines in.
• ◆Point-slope form, y - y1 = m(x - x1). It says "this line has slope m and goes through the point (x1, y1)". Nothing has been tidied away, which is exactly why it is the easiest form to *write down* when those are the two facts you were given.
• ◆Standard form, Ax + By = C. Both variables sit on the left with whole-number coefficients. It treats x and y evenhandedly, which makes it the form that can describe a vertical line, and the form that systems of equations like best.

They all describe the same set of points. y = 2x - 1, y - 5 = 2(x - 3) and 2x - y = 1 are three spellings of one line, and the skill worth having is moving between them without losing a sign on the way.

Point-slope form is not something to memorise. It falls out of the definition of slope in one step.

Slope means rise over run. Take a known point (x1, y1) on the line and let (x, y) be any other point on it. Then

m = (y - y1) / (x - x1)

Multiply both sides by (x - x1) and you have

y - y1 = m(x - x1)

That is the whole derivation. It also explains the minus signs, which are where most of the errors live: the form subtracts the coordinates of the known point, so a point with a negative coordinate produces a double negative. Through (-3, 4) with slope 2 the honest first line is

y - 4 = 2(x - (-3))

and only then does it tidy to y - 4 = 2(x + 3). Writing the brackets before simplifying them costs a second and saves the sign.

Given two points instead of a slope, nothing changes — you just compute m from the two of them first, then use either point. Both choices give the same line, so pick whichever has the friendlier numbers.

Point-slope to slope-intercept. Multiply out the bracket and move the constant across. From y - 4 = 2(x + 3): expand to y - 4 = 2x + 6, then add 4 to both sides to get y = 2x + 10.

Slope-intercept to standard. Two moves, in this order.

• ◆Clear the fractions: multiply *every* term by the lowest common denominator. From y = (3/4)x - 2, multiplying by 4 gives 4y = 3x - 8.
• ◆Carry the x-term to the left, changing its sign: -3x + 4y = -8. Standard form is usually written with A greater than 0, so multiply the whole equation by -1: 3x - 4y = 8.

Standard to slope-intercept. Get the y-term alone and divide. From 3x - 4y = 8: -4y = -3x + 8, then y = (3/4)x - 2. Doing that once in general gives a shortcut worth keeping: the line Ax + By = C has slope -A/B. For 3x - 4y = 8 that is -3/(-4) = 3/4, which agrees.

If the equation arrives out of order, such as 4y - 3x = 8 or 4y = 8 + 3x, put it back into Ax + By = C first. Tidy, then read.

An intercept is where a line meets an axis, and each axis is defined by one coordinate being zero. That single sentence is the whole method.

• ◆Every point on the x-axis has y = 0. So to find the x-intercept, set y = 0 and solve for x.
• ◆Every point on the y-axis has x = 0. So to find the y-intercept, set x = 0 and solve for y.

Standard form makes this fast, because setting a variable to zero deletes a whole term. For 3x + 5y = 30: set y = 0 to get 3x = 30, so x = 10; set x = 0 to get 5y = 30, so y = 6. The line crosses at (10, 0) and (0, 6). Notice which is which — the x-intercept is a point on the *x*-axis, so its y-coordinate is 0, not the other way round.

Two lines have no slope-intercept form at all, and standard form handles both. A horizontal line has slope 0, so y = mx + b collapses to y = b: every point sits at the same height. A vertical line has a run of 0 between any two of its points, so its slope cannot be worked out at all; its equation is x = k, and it never appears as y = something.

Two lines are parallel when they have the same slope and different intercepts — same steepness, different starting height, so they never meet.

Two lines are perpendicular when their slopes multiply to -1. In practice that means the new slope is the negative reciprocal of the old one: turn the fraction over and change its sign. The perpendicular partner of 2/3 is -3/2; of 4 (that is, 4/1) it is -1/4; of -5 it is 1/5. Both halves matter, and forgetting the sign is the single most common slip in this topic.

So "the line through (6, 1) perpendicular to y = (2/3)x + 5" is two short steps: the new slope is -3/2, and then point-slope with (6, 1) finishes it.

This is the last purely linear skill, and three later topics lean on it directly. Systems of equations ask where two of these lines meet, and standard form is the shape that elimination wants. Linear models read m as a rate of change and b as a starting amount, so writing a line from two data points is writing a prediction. And the formula for the distance from a point to a line is stated for Ax + By = C, which is one more reason standard form is worth being fluent in.