Skill: Write equations of lines in standard form

Explanation and Free Practice Resources

A line in standard form is represented as $Ax+By=C$ , typically with integer coefficients, $A\ge 0$ , and no common factor, so that equivalent equations describe the same set of points. The understanding includes converting from slope-intercept or point-slope form, determining an equation from a graph or given points, and recognizing horizontal and vertical lines; it is limited to linear equations in two variables and does not include parametric, vector, or higher-dimensional forms.

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Detailed Explanation: Write equations of lines in standard form

Standard form for a linear equation is

Ax+By=C,

where $A$ , $B$ , and $C$ are integers, $A\ge 0$ , and the coefficients have no common factor other than $1$ .

To convert an equation to standard form:

Clear any fractions by multiplying every term by the least common denominator.
Move the $x$ - and $y$ -terms to the same side.
Arrange the terms as $Ax+By=C$ .
Make sure $A$ is positive and simplify if all coefficients share a common factor.

Example: Write the equation

y=\frac{2}{3}x-4

in standard form.

First, clear the fraction by multiplying every term by $3$ :

3y=2x-12.

Move the $x$ -term to the left side:

-2x+3y=-12.

Standard form requires $A\ge 0$ . Since the coefficient of $x$ is negative, multiply the entire equation by $-1$ :

2x-3y=12.

Therefore, the equation in standard form is

\boxed{2x-3y=12}.

Multiplying every term by the same nonzero number does not change the line, so the new equation represents the same set of points.

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