Chapter 4 Linear Functions
4.1 Linear Equations in Two Variables
Learning Objectives
In this section, you will:
- Write a linear equation in two variables
- Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
- Write the equation of a line parallel or perpendicular to a given line.
Write a Linear Equation in Two Variables
Perhaps the most familiar form of a linear equation is the slope-intercept form, written as[latex]\,y=mx+b,[/latex] where[latex]\,m=\text{slope}\,[/latex]and[latex]\,b=y\text{−intercept}\text{.}\,[/latex]Let us begin with the slope.
The Slope of a Line
The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.
The Slope of a Line
The slope of a line, m, represents the change in y over the change in x. Given two points,[latex]\,\left({x}_{1},{y}_{1}\right)\,[/latex]and[latex]\,\left({x}_{2},{y}_{2}\right),[/latex] the following formula determines the slope of a line containing these points:
Finding the Slope of a Line Given Two Points
Find the slope of a line that passes through the points[latex]\,\left(2,-1\right)\,[/latex]and[latex]\,\left(-5,3\right).[/latex]
Show Solution
We substitute the y-values and the x-values into the formula.
The slope is[latex]\,-\frac{4}{7}.[/latex]
Analysis
It does not matter which point is called[latex]\,\left({x}_{1},{y}_{1}\right)\,[/latex]or[latex]\,\left({x}_{2},{y}_{2}\right).\,[/latex]As long as we are consistent with the order of the y terms and the order of the x terms in the numerator and denominator, the calculation will yield the same result.
Try It
Find the slope of the line that passes through the points[latex]\,\left(-2,6\right)\,[/latex]and[latex]\,\left(1,4\right).[/latex]
Show Solution
[latex]m=-\frac{2}{3}[/latex]
Identifying the Slope and y-intercept of a Line Given an Equation
Identify the slope and y-intercept, given the equation[latex]\,y=-\frac{3}{4}x-4.[/latex]
Show Solution
As the line is in[latex]\,y=mx+b\,[/latex]form, the given line has a slope of[latex]\,m=-\frac{3}{4}.\,[/latex]The y-intercept is[latex]\,b=-4.[/latex]
Analysis
The y-intercept is the point at which the line crosses the y-axis. On the y-axis,[latex]\,x=0.\,[/latex]We can always identify the y-intercept when the line is in slope-intercept form, as it will always equal b. Or, just substitute[latex]\,x=0\,[/latex]and solve for y.
The Point-Slope Formula
Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.
This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and wr