Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
$ax + by = c$ is the general equation of a linear equation in two variables, $x$ and $y$. This equation represents a straight line in the coordinate plane, where $a$, $b$, and $c$ are real numbers that define the slope and y-intercept of the line.
5 Must Know Facts For Your Next Test
The coefficients $a$, $b$, and $c$ in the equation $ax + by = c$ define the characteristics of the linear equation, such as the slope and y-intercept.
The graph of $ax + by = c$ is a straight line in the coordinate plane, and the line passes through the point $(-c/a, 0)$ on the $x$-axis and the point $(0, -c/b)$ on the $y$-axis.
The slope of the line represented by $ax + by = c$ is $-a/b$, and the y-intercept is $c/b$.
The general equation $ax + by = c$ can be transformed into the slope-intercept form $y = mx + b$ by solving for $y$.
The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, can be derived from the general equation $ax + by = c$ by using a known point $(x_1, y_1)$ on the line.
Review Questions
Explain how the coefficients $a$, $b$, and $c$ in the equation $ax + by = c$ define the characteristics of the linear equation.
The coefficients $a$, $b$, and $c$ in the equation $ax + by = c$ play a crucial role in defining the characteristics of the linear equation. The coefficient $a$ represents the slope of the line, the coefficient $b$ represents the y-intercept, and the constant $c$ determines the point where the line intersects the $x$-axis. By knowing these three values, you can determine important features of the line, such as its orientation, direction, and location in the coordinate plane.
Describe how the general equation $ax + by = c$ can be transformed into the slope-intercept form $y = mx + b$.
To transform the general equation $ax + by = c$ into the slope-intercept form $y = mx + b$, you need to solve the equation for $y$. This can be done by first rearranging the terms to isolate $y$ on one side of the equation: $by = -ax + c$. Then, dividing both sides by $b$ to get the slope-intercept form: $y = (-a/b)x + (c/b)$. In this form, the slope is $-a/b$ and the y-intercept is $c/b$, which provides a more intuitive representation of the linear equation.
Explain how the general equation $ax + by = c$ can be used to determine the relationship between two lines, such as whether they are parallel or perpendicular.
The general equation $ax + by = c$ can be used to determine the relationship between two lines, such as whether they are parallel or perpendicular. If two lines have the same slope, meaning the coefficients $a$ and $b$ are the same for both lines, then the lines are parallel. Conversely, if the slopes of the two lines are negative reciprocals, meaning $a_1/b_1 = -a_2/b_2$, then the lines are perpendicular. This relationship can be derived directly from the general equation, which allows you to analyze the relative orientation of lines in the coordinate plane.
The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This form can be derived from the general equation $ax + by = c$ by solving for $y$.
The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope. This form can also be derived from the general equation $ax + by = c$.
Parallel and Perpendicular Lines: The general equation $ax + by = c$ can be used to determine the relationship between two lines. If two lines have the same slope, they are parallel, and if the slopes are negative reciprocals, the lines are perpendicular.