Back to the on-screen lesson ·
Slope-intercept, point-slope and standard forms, lines through two points, and the slope rules for parallel and perpendicular lines.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn that one straight line can be written three ways — $y = mx + b$, $y - y_1 = m(x - x_1)$ and $Ax + By = C$ — and how to move between them depending on what you are given. You then meet the two slope rules that decide how two lines sit relative to each other: equal slopes for parallel, and slopes multiplying to $-1$ for perpendicular.
You already read $y = mx + b$: $m$ is the slope and $b$ is where the line crosses the $y$-axis. You already find a slope as rise over run. This lesson adds two more ways of writing the same line and shows that all three carry the same information, differently arranged.
Slope-intercept form: $y = mx + b$. Shows the slope and the $y$-intercept at a glance.
Point-slope form: $y - y_1 = m(x - x_1)$. Built straight from a slope and any one point on the line.
Standard form: $Ax + By = C$. Convenient for intercepts and for systems; the slope is not $A$.
Intercept: where the line crosses an axis. The $y$-intercept has $x = 0$; the $x$-intercept has $y = 0$.
A straight line is fixed by two numbers, and each form displays them differently. Slope-intercept $y = mx + b$ shows the slope and the crossing point of the $y$-axis, and is the form to aim for when you want to graph or compare. Point-slope $y - y_1 = m(x - x_1)$ is what you write the instant you know a slope and one point, with no rearranging at all. Standard form $Ax + By = C$ hides the slope but makes both intercepts easy: set $x = 0$ for one, $y = 0$ for the other. Given two points, always start with the slope, $m = \frac{y_2 - y_1}{x_2 - x_1}$, and then use either point to pin the rest down. Two special cases sit outside the pattern: $y = c$ is horizontal with slope $0$, and $x = c$ is vertical and has no slope at all, because its run is zero.
Another way: diagram
One line drawn on a grid three times over, labelled $y = 3x - 1$, $y - 2 = 3(x - 1)$ and $3x - y = 1$, with the point $(1, 2)$ and the intercept $(0, -1)$ marked on it.
Another way: story
The three forms are three ways of describing the same journey: 'start here and climb at this rate', 'pass through this landmark at this rate', and 'these are the two places you cross the axes'. Convert to whichever one answers the question asked.
"In $6x - 2y = 8$ the slope is $6$." Not until it is solved for $y$. $-2y = -6x + 8$ gives $y = 3x - 4$, so the slope is $3$.
"Point-slope form needs the $y$-intercept." It needs any point. That is the whole reason it exists: you often know a point on the line and its slope, and not the intercept.
"Slope is run over rise." It is rise over run — the change in $y$ divided by the change in $x$. Getting it upside down turns every steep line into a flat one.
Slope: $\frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$.
Rise over run, in that order.
Use $(1, 2)$: $2 = 3(1) + b$, so $b = -1$.
Either point works.
The line is $y = 3x - 1$.
Solve for $y$: $-2y = -6x + 8$.
Move the $x$ term across.
Divide by $-2$: $y = 3x - 4$.
Every term changes sign.
So the slope is $3$ and the $y$-intercept is $-4$.
On the $y$-axis, $x = 0$, so the equation becomes $3y = 12$.
So $y = 4$: the line crosses at $(0, 4)$.
A line has slope $5$ and $y$-intercept $-6$. Write $y$ in terms of $x$.
Answer:
Write the equation of the line through $(3, 13)$ and $(7, 37)$ as $y$ in terms of $x$.
Answer:
What is the slope of $12x - 3y = 21$?
answer
Where does the line $3x + 2y = -4$ cross the $y$-axis? Give the $y$ value.
Answer:
You already know that the slope is how steep a line is, and you have just practised reading one off an equation. Everything here is two short rules about slopes — plus the reciprocal, which you met with negative exponents: the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.
Parallel: two lines that never meet. Written $\parallel$.
Perpendicular: two lines that meet at a right angle. Written $\perp$.
Reciprocal: the number turned over. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$; the reciprocal of $4$ is $\frac{1}{4}$.
Negative reciprocal: turned over and with its sign changed. The negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$.
Two lines are parallel exactly when their slopes are equal — they climb at the same rate, so the gap between them never closes — and they are different lines only if their intercepts differ. Two lines are perpendicular exactly when their slopes multiply to $-1$, which is the same as saying each slope is the other turned over with its sign changed: $\frac{2}{3}$ pairs with $-\frac{3}{2}$, and $4$ pairs with $-\frac{1}{4}$. The reason is visible in a picture: turning a line a quarter turn swaps its rise with its run and reverses one of them, which is precisely what taking the negative reciprocal does. To build such a line through a given point, work out the required slope first and then use point-slope form.
Another way: diagram
A right triangle with legs $3$ across and $2$ up drawn beside the same triangle rotated a quarter turn, now $2$ across and $3$ down, showing slope $\frac{2}{3}$ becoming $-\frac{3}{2}$.
Another way: story
Parallel is 'same steepness'. Perpendicular is 'swap rise and run, then flip one sign'. Everything else — writing the equation, using a point — is work you have already done.
"Perpendicular slopes are just negatives of each other." $2$ and $-2$ are not perpendicular: their product is $-4$. The perpendicular partner of $2$ is $-\frac{1}{2}$.
"Same equation, so parallel." Lines with the same slope and the same intercept are the same line, not two parallel ones. Parallel needs equal slopes and different intercepts.
"A vertical and a horizontal line break the rule." They are perpendicular, but the product test cannot be applied: a vertical line has no slope, so the rule is stated for lines that have one.
The perpendicular slope is the negative reciprocal of $2$: $-\frac{1}{2}$.
Turn over, change sign.
The point $(0, 5)$ is on the $y$-axis, so the intercept is $5$.
$x = 0$ gives the intercept directly.
The line is $y = -\frac{1}{2}x + 5$.
Multiply them: $\frac{2}{3} \times \left(-\frac{3}{2}\right)$.
The test is the product.
The $2$s and the $3$s cancel, leaving $-1$.
A product of $-1$ means yes, they are perpendicular.
Solve for $y$: $y = -3x + 2$, so the slope is $-3$.
A parallel line has the same slope, $-3$.
What is the slope of a line parallel to $y = -3x + 9$?
Answer:
What is the slope of a line perpendicular to $y = 6x + 1$? Give a fraction.
Answer:
Write the line perpendicular to $y = -\dfrac{1}{9}x - 4$ that passes through $(0, -9)$, as $y$ in terms of $x$.
Answer:
One line has slope $\dfrac{2}{5}$ and another has slope $-\dfrac{5}{2}$. Are they perpendicular?
A garden path runs in a straight line from $(0, 0)$ to $(4, 5)$. A second path runs from $(4, 5)$ to $(-1, 9)$. Do the two paths meet at a right angle where they join?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Write the line with slope $-7$ through $(6, -6)$ in point-slope form. Fill in both blanks.
$y - $ k $ = -7(x - $ h $)$
A line parallel to $y = 8x$ passes through $(-2, -23)$. What is its $y$-intercept?
Answer:
You can write a line in any form and use the slope rules for parallel and perpendicular lines. Explain why the slope of $6x - 2y = 8$ is $3$ and not $6$, and why the perpendicular partner of slope $2$ is $-\frac{1}{2}$ rather than $-2$.
8. Your turn: where does $2x + 3y = 12$ cross the $y$-axis?, step 2
19. Your turn: what slope is parallel to $3x + y = 2$?, step 2