Standard Form of a Linear Equation – Ax + By = C (Algebra 1)

Contents

Key takeaway: The standard form of a linear equation is Ax + By = C, where A, B and C are integers, A is not negative, and A and B are not both zero.

Grade 9 – Algebra 1A-CED.A.2Functions and Graphs

Definition of Standard Form of a Linear Equation

Definition

The standard form of a linear equation puts both variable terms on the left and the constant on the right. It is popular for problems that split a fixed total between two items, such as two prices that add up to a budget.

Its main strength is how quickly the intercepts appear. Cover the $y$-term to find the $x$-intercept and cover the $x$-term to find the $y$-intercept. Its drawback is that the slope is hidden: you have to rearrange to see it. Most teachers also ask for no fractions or decimals in $A$, $B$ and $C$, and for $A$ to be positive.

Formula and notation

The standard form of a linear equation follows the pattern

\[Ax + By = C\]

The slope is $-\frac{A}{B}$, the $x$-intercept is $(\frac{C}{A}, 0)$ and the $y$-intercept is $(0, \frac{C}{B})$ when $A$ and $B$ are not zero.

Cover one term to read each intercept: 60 ÷ 4 = 15 and 60 ÷ 3 = 20.

Key facts

Standard form needs integer coefficients and a nonnegative A, with no fractions.
Set x = 0 or y = 0 to read off the intercepts in one step.
The slope is negative A over B, so it is not visible at a glance.
In standard form, horizontal lines have A = 0 and vertical lines have B = 0.

Standard Form of a Linear Equation example

Problem

A booth sells popcorn for \$4 and drinks for \$3. The sales target is \$60, so $4p + 3d = 60$. Find both intercepts and rewrite the equation in slope-intercept form.

Let $d = 0$: $4p = 60$, so $p = 15$. The intercept is $(15, 0)$.
Let $p = 0$: $3d = 60$, so $d = 20$. The intercept is $(0, 20)$.
Solve for $d$: $3d = -4p + 60$, so $d = -\frac{4}{3}p + 20$.

The intercepts are $(15, 0)$ and $(0, 20)$, and the slope is $-\frac{4}{3}$.

Common mistake

Watch out

Converting $y = -\frac{2}{3}x + 4$, Tessa writes $2x – 3y = 12$.

Why it is wrong

Multiply by 3 first: $3y = -2x + 12$. Then add $2x$ to both sides, which gives $2x + 3y = 12$. Her minus sign came from moving $-2x$ the wrong way. Check with the point $(0, 4)$: $2(0) + 3(4) = 12$.

Related terms and lessons

Algebra 1 · Linear Functions: Slope-Intercept, Point-Slope and Standard FormLesson Worksheet Quiz

Browse the glossary: A–Z list · Algebra 1 vocabulary · Functions and Graphs

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