# Sections 2.4 & 2.5 - Lines and Linear Equations in Two Variables

Section Objectives

1. Determine solutions of two-variable linear equations.
2. Graph a line by finding two points on the line.
3. Find the $x$- and $y$-intercepts of a line.
4. Compute the slope of a line and interpret it as a rate of change.
5. Identify equations of horizontal or vertical lines and graph them.

### Equations of Lines

Any linear equation in two variables can be written in the form

$Ax+By=C,$

where $A$ and $B$ are not both zero. The form $Ax+By=C$ is called standard form.

A solution of a two-variable linear equation is a point $(x,y)$ that satisfies the equation. Every two-variable linear equation has infinitely many solutions. If all such solutions are plotted in a rectangular coordinate system, the resulting graph is a line.

#### Examples

• Show that both $(1,1)$ and $(1/3,2)$ are solutions of $3x+2y=5$.

• Show that $(0,5)$ and $(-3,5)$ are solutions of $2y=10$. Can you find two additional solutions?

• Find two solutions of $2x-y=9$.

### Graphing Lines

Remember from geometry that a line is uniquely determined by two distinct points. Therefore, one approach (among others that we will learn) to graphing the line described by a two-variable linear equation is to:

1. determine two solutions (i.e, two points on the line),
2. plot them, and
3. sketch a line through them.

Some of us may also like to find one more solution to check our work.

#### Examples

• Graph the line described by $y-4x=1$.

• Graph the line described by $3x-2y=6$.

• Graph the line described by $x=-3$. What word would you use to describe the line?

• Graph the line described by $y=2$. What word would you use to describe the line?

### Intercepts

Points of the form $(a,0)$ and $(0,b)$ are called $x$- and $y$-intercepts, respectively. These are points where a graph intersects the $x$-axis or the $y$-axis.

In problem situations, Intercepts often have some kind of physical significance. They are usually worth finding! And they can be useful for graphing.

#### Examples

• Graph the line whose $x$-intercept is $(6,0)$ and whose $y$-intercept is $(0,2)$.

• Determine the $x$- and $y$-intercepts. Then use them to graph the line. $-2x+4y=12$

### Slope

The slant of a line is measured by a number called slope. Before we define slope, here are three important ideas to keep in mind:

1. Slope is only defined for non-vertical lines.
2. The slope of a line is the same at any point on the line.
3. The slope is a measure of the rate of change of $y$ with respect to $x$.

Suppose $(x_1, y_1)$ and $(x_2,y_2)$ are any two distinct points on a non-vertical line. The slope of the line is given by $\quad \displaystyle m = \frac{y_2-y_1}{x_2-x_1} = \frac{\Delta y}{\Delta x}$.

#### Examples

• Determine the slope of the line that passes through the two points $(4,8)$ and $(-1,-7)$.

• Determine two points on the line described by the equation $x-3y=9$. Then use your points to find the slope of the line.

### Parallel and Perpendicular Lines

Two non-vertical lines are parallel if and only if they have the same slope.

Two non-vertical lines are perpendicular if and only if their slopes are opposite reciprocals (that is, the product of their slopes is $-1$).

What if one or both of the lines are vertical?

#### Examples

• The line $L$ passes through the points $(4,6)$ and $(-2,5)$. Find the slope of a line parallel to $L$. Find the slope of a line perpendicular to $L$.

• The line $V$ passes through the points $(0,3)$ and $(-3,3)$. Find an equation of a line parallel to $V$. Find an equation of a line perpendicular to $V$.

#### Miscellaneous Examples

• Write equations for the horizontal and vertical lines through $(-4,7)$.

• Sal fixes vintage arcade games. He charges a flat fee of $140 to make a house call, but then he charges a constant hourly rate on top of that. He recently made a house call to fix a Centipede game and ended up billing the a client$230 after 2 hours of work. Sketch the graph the shows how much Sal makes versus time (in hours). What does the slope of the graph represent?