Section 4.5: Graphing Linear Equations in Standard Form

Property

An equation of the form $Ax + By = C$, where $A$ and $B$ are not both zero, is called a linear equation in two variables.
A linear equation is in standard form when it is written $Ax + By = C$, where $A$, $B$, and $C$ are real numbers.

Examples

Property

We can write the equation of any non-vertical line in slope-intercept form by solving the equation for $y$ in terms of $x$.

Caution: Do not confuse solving for $y$ with finding the $y$-intercept. When we solve for $y$, we are writing the equation in another form, so both variables, $x$ and $y$, still appear in the equation.

Examples

• To convert $6x + 3y = 12$, subtract $6x$ from both sides to get $3y = -6x + 12$. Then divide all terms by $3$ to get the final form $y = -2x + 4$.
• For the equation $5x - 2y = 10$, subtract $5x$ to get $-2y = -5x + 10$. Divide everything by $-2$ to find the slope-intercept form, $y = \frac{5}{2}x - 5$.
• To solve $x + 4y = 8$ for $y$, subtract $x$ from both sides giving $4y = -x + 8$. Then divide by $4$ to get $y = -\frac{1}{4}x + 2$.