Graphing Linear Equations and Finding Midpoints in the Coordinate Plane

Linear Equations in Two Variables

Standard Form of a Linear Equation

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

• Standard form: $Ax + By = C$

• Variables: $x$ and $y$

• Coefficients: $A$, $B$ (not both zero)

Intercepts of a Line

Two useful points for graphing are the x-intercept and y-intercept:

• x-intercept: The point where the line crosses the x-axis ($y = 0$).

• y-intercept: The point where the line crosses the y-axis ($x = 0$).

To find the intercepts, substitute 0 for the other variable and solve:

• To find the x-intercept, let $y = 0$ and solve for $x$.

• To find the y-intercept, let $x = 0$ and solve for $y$.

Example: Find the x- and y-intercepts and Graph the Equation $2x - y = 4$

• Find the x-intercept: Set $y = 0$:

$\begin{align*} 2x - (0) &= 4 \\ 2x &= 4 \\ x &= \frac{4}{2} = 2 \end{align*}$

The x-intercept is $(2, 0)$.

• Find the y-intercept: Set $x = 0$:

$\begin{align*} 2(0) - y &= 4 \\ 0 - y &= 4 \\ -y &= 4 \\ y &= -4 \end{align*}$

The y-intercept is $(0, -4)$.

Graphing the Line

Plot the intercepts $(2, 0)$ and $(0, -4)$ on the coordinate plane, then draw a straight line through these points.

Special Cases: Horizontal and Vertical Lines

Equations with One Variable

If the equation is missing a variable, it represents a special case:

• Horizontal line: $y = k$ (where $k$ is a constant)

• Vertical line: $x = h$ (where $h$ is a constant)

Example: Graph the Horizontal Line $y = 3$

• This line passes through all points with $y$-coordinate 3.

Example: Graph the Vertical Line $x = -2$

• This line passes through all points with $x$-coordinate -2.

Graphing by Slope-Intercept Form

Slope-Intercept Form

The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope ($m$): The rate of change, or how much $y$ increases or decreases as $x$ increases by 1.

• Y-intercept ($b$): The value of $y$ when $x = 0$.

To graph a line in standard form, you can rewrite it in slope-intercept form or use a t-table to plot points.

Lines Passing Through the Origin

If a linear equation in standard form has $C = 0$, the line passes through the origin $(0, 0)$.

• Example: $3x - y = 0$ or $-x = y$

Midpoint of a Line Segment

Definition and Formula

The midpoint of a line segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$ is the point halfway between them. The formula is:

$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Example: Find the Midpoint of $P(-5, 8)$ and $Q(3, -2)$

• Calculate the average of the x-coordinates: $\frac{-5 + 3}{2} = -1$

• Calculate the average of the y-coordinates: $\frac{8 + (-2)}{2} = 3$

The midpoint is $(-1, 3)$.

Slope of a Line

Slope Formula

The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Example: Find the Slope of the Line through $(-6, 6)$ and $(4, 2)$

$m = \frac{2 - 6}{4 - (-6)} = \frac{-4}{10} = -\frac{2}{5}$

Example: Find the Slope of the Line $5x - 2y = 14$

• Rewrite in slope-intercept form:

$\begin{align*} 5x - 2y &= 14 \\ -2y &= -5x + 14 \\ y &= \frac{5}{2}x - 7 \end{align*}$

The slope is $\frac{5}{2}$.

Summary Table: Types of Lines and Their Equations

Type of Line	Equation	Slope
General (oblique)	$y = mx + b$	$m$ (finite)
Horizontal	$y = k$	$0$
Vertical	$x = h$	Undefined
Through origin	$y = mx$ or $Ax + By = 0$	$m$ (finite)

Additional info: This table summarizes the main types of lines encountered in algebra and their properties.