Standard Form: Ax + By = C
Write linear equations in standard form and use it to find x- and y-intercepts easily. Convert between all three forms (slope-intercept, point-slope, standard) and recognize the advantages of each.
A lesson of Algebra 1, in Linear Functions, about 30 minutes.
Standard Form
By the end of this lesson, you will be able to recognize standard form, find both intercepts from it quickly, convert to and from slope-intercept form, and say which situations it is the natural way to write.
The Form That Treats Both Variables Equally
$$Ax + By = C$$
Slope-intercept singles out $y$: it says "here is how to compute $y$ from $x$". Standard form does not. It states a relationship between the two, with neither one the subject.
$$3x + 2y = 12$$
That reads: three of one thing plus two of another comes to twelve. No variable is the answer and no variable is the input. And that is exactly how constraints arrive in the world.
Adult tickets cost $$8$, child tickets cost $$5$, and the takings were $$240$.
$$8a + 5c = 240$$
Writing that as $c = \dots$ would be perverse. Neither number depends on the other; they are tied together by a total, and standard form is the shape of a total.
By convention $A$, $B$ and $C$ are integers and $A$ is not negative, so $-2x + 3y = 6$ is usually written $2x - 3y = -6$.
What It Is Good At: Both Intercepts
The reason to keep this form is that both intercepts fall out in one step each.
$$3x + 2y = 12$$
$x$-intercept: set $y = 0$. Then $3x = 12$, so $x = 4$. Crosses at $(4, 0)$.
$y$-intercept: set $x = 0$. Then $2y = 12$, so $y = 6$. Crosses at $(0, 6)$.
Plot those two and the line is drawn. This is the fastest graphing method there is for an equation in this form, and it needs no rearranging at all.
What it is bad at: the slope is not visible. It can be computed, and there is a formula, but reading it off the way you read $m$ from $y = mx + b$ is not available. Use the right form for the question.
Example 1: Graphing by Intercepts
Here is the whole method on one equation, worked all the way through. $$4x + 3y = 24$$
- Set y to zero and write what is left 4x = 24
- Solve it. What is the x-intercept? 6
- Now set x to zero and write what is left 3y = 24
- Solve it. What is the y-intercept? 8
Plot It: The Two Intercepts
Worked Example 1: Your Turn on Intercepts
$$5x - 2y = 20$$
Converting Between the Forms
Standard to slope-intercept: solve for $y$.
$$3x + 2y = 12 \longrightarrow 2y = -3x + 12 \longrightarrow y = -\frac{3}{2}x + 6$$
Slope-intercept to standard: clear fractions, then move the $x$ term across.
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