Standard Form of a Linear Equation

Sommaire

In short: The standard form of a linear equation is Ax + By = C, with integers A, B and C, where x and y sit on the left and a constant sits on the right.

Algebra 1Functions and GraphsA-CED.A.2

What is Standard Form of a Linear Equation?

The standard form of a linear equation collects both variables on one side: $Ax + By = C$. By convention A, B and C are integers with no common factor, and A is not negative.

This layout is handy whenever two quantities share a budget, like tickets and snacks bought with $60. Its intercepts are easy to find: cover the y term to get $x = \frac{C}{A}$ at $y = 0$, and cover the x term to get $y = \frac{C}{B}$ at $x = 0$.

How to spot it

An equation shows x and y terms added together on one side and a number on the other.
A word problem splits a fixed total between two items with different prices.
You are asked to find the intercepts quickly to graph a line.
A teacher wants integer coefficients and no fractions in the final equation.

The method

Rewrite into the standard form of a linear equation: x term and y term on the left, the constant on the right.
Clear fractions by multiplying every term by a common denominator.
If A is negative, multiply the whole equation by $-1$; divide out any common factor.
To graph, set $y = 0$ for the x-intercept, set $x = 0$ for the y-intercept, and join the two points.

3x + 4y = 24 meets the axes at (8, 0) and (0, 6).

Standard Form of a Linear Equation: solved example

Problem

Dara has $60 for a school fair. Tickets cost $4 each and drinks cost $6 each. Write an equation in standard form, find both intercepts, and say how many tickets she can buy if she gets 6 drinks.

Solution

With x tickets and y drinks, $4x + 6y = 60$, which divides by 2 to $2x + 3y = 30$.

Intercepts: $y = 0$ gives $x = 15$ (all tickets); $x = 0$ gives $y = 10$ (all drinks). With 6 drinks: $4x + 36 = 60$, so $x = 6$. She can buy 6 tickets.

Try it

Question 1

Find both intercepts of $5x – 2y = 20$.

Show the answer

$y = 0$: $x = 4$, so $(4, 0)$. $x = 0$: $y = -10$, so $(0, -10)$.

Question 2

Write $y = -\frac{2}{3}x + 4$ in standard form with integer coefficients.

Show the answer

Multiply by 3: $3y = -2x + 12$. Then $2x + 3y = 12$.

Standard Form of a Linear Equation vs. Slope-Intercept Form

The slope-intercept layout isolates y so that the slope and the y-intercept can be read off at once. The standard layout keeps both variables together on the left, which suits budgets and intercepts but hides the slope; to see it, divide through by B and solve for y.

Slope-Intercept Form

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