Straight-Line Equations

Introduction

Basic Competencies

3.4 Analyze linear functions as straight-line equations and interpret their graphs in contextual problems.

4.4 Solve contextual problems related to linear functions as straight-line equations.

Learning Objectives

After studying this material, students are expected to be able to:

1. analyze linear functions and their graphs,
2. determine the slope of a line,
3. determine the relationship between two lines from their slopes,
4. form a straight-line equation from the given information.

Material

1. Graph of a Straight-Line Equation

Observe the following graph.

For the equations $y = 2x$ and $y = 2x + 1$, the coefficient of $x$ is 2. This value describes the comparison between the change in $y$ and the change in $x$. It is called the slope or gradient of the line.

If the slope is denoted by $m$, then the general form of a straight-line equation is:

where:

• $m$ = slope,
• $c$ = constant.

Examples of straight-line equations:

1. $y = 3x - 4$
2. $2x - 3y - 5 = 0$

To draw the graph of $y = mx + c$:

1. find the intercept with the $x$-axis,
2. find the intercept with the $y$-axis,
3. connect the two points.

Example

Draw the graph of:

Find the $x$-intercept by setting $y = 0$:

So the $x$-intercept is $(2,0)$.

Find the $y$-intercept by setting $x = 0$:

So the $y$-intercept is $(0,6)$.

2. Determining the Slope of a Straight Line

The slope shows the degree of inclination of a line relative to the $x$-axis and is usually denoted by $m$.

A. Slope of a line through the origin $(0,0)$ and a point $(x_1,y_1)$

Formula:

B. Slope of a line crossing the $x$-axis at $(b,0)$ and the $y$-axis at $(0,a)$

Formula:

C. Slope of a line through two points $(x_1,y_1)$ and $(x_2,y_2)$

Formula:

Example

Find the slope of the line through $A(-1,3)$ and $B(7,-5)$.

Slope Simulation

Use the following simulation to better understand slope visually.

3. Relationship Between Two Lines from Their Slopes

A. Parallel lines

Two parallel lines have the same slope.

Example:

1. The line through $(1,6)$ and $(3,2)$
2. The line through $(2,-9)$ and $(-3,1)$

First slope:

Second slope:

Since $m_1 = m_2$, the lines are parallel.

B. Perpendicular lines

Two lines are perpendicular if:

Example:

1. The line through $(0,-8)$ and $(3,1)$
2. The line through $(-3,3)$ and $(6,0)$

First slope:

Second slope:

Because:

the lines are perpendicular.

4. Forming a Straight-Line Equation

A. Through one point with slope $m$

If a line passes through $(x_1,y_1)$ and has slope $m$, then:

Example

Find the equation of a line with slope 3 passing through $(-2,5)$.

So the equation is:

B. Through two points

If a line passes through $(x_1,y_1)$ and $(x_2,y_2)$, then:

or:

Example

Find the equation of the line through $(10,3)$ and $(2,7)$.

So the equation is:

Evaluation

Answer the following questions to test your understanding.

1. Find the slope of the line in the figure below.

2. The line through $P(2,y)$ and $Q(5,7)$ has slope $-1$. Find $y$.
3. Line $g$ is perpendicular to the line through $P(0,7)$ and $Q(2,1)$. Find the slope of line $g$.
4. Find the equation of the line through $(1,2)$ and perpendicular to $3x+4y=8$.
5. Find the equation of a line with slope $\frac{2}{3}$ passing through $(-6,-5)$.