3.4 Graph of Linear Equations. Objective 1. Use the slope-intercept form of the equation of a line. Slide 3.4-3. Use the slope-intercept form of the equation of line.
Use the slope-intercept form of the equation of a line.
In Section 3.3, we found the slope of a line by solving for y. In that form, the slope is the coefficient of x. For, example, the slope of the line with equation y = 2x + 3 is 2. So, what does 3 represent?
Suppose a line has a slope m and y-intercept (0,b). We can find an equation of this line by choosing another point (x,y) on the line as shown. Then we use the slope formula.
Change in y-values
Change in x-values
Subtract in the denominator.
Multiply by x.
Add b to both sides.
The result is the slope-intercept form of the equation of a line, because both the slope and the y-intercept of the line can be read directly from the equation. For the line with the equation y = 2x + 3, the number 3 gives the y-intercept (0,3).
The slope-intercept form of the equation of a line with slope m and y-intercept (0,b) is
Where m is the slope and b is the y-intercept (0,b).
y-intercept: (0,− 6)
CLASSROOM EXAMPLE 1
Identifying Slopes and y-Intercepts
Identify the slope and y-intercept of the line with each equation.
Slope: − 1
Writing an Equation of a Line
Write an equation of the line with slope −1 and y-intercept (0,5).
Graph a line by using its slope and a point on the line.
Graphing a Line by Using the Slope and y-Intercept
Step 1:Write the equation in slope-intercept form, if necessary, by solving for y.
Step 2:Identify the y-intercept. Graph the point (0,b).
Step 3:Identify slope m of the line. Use the geometric interpretation of slope (“rise over run”) to find another point on the graph by counting from the y-intercept.
Step 4:Join the two points with a line to obtain the graph.
CLASSROOM EXAMPLE 3
Graphing Lines by Using Slopes and y-Intercepts
Graph 3x – 4y = 8 by using the slope and y-intercept.
Graph the line through (2,−3) with slope
CLASSROOM EXAMPLE 4
Graphing a Line by Using the Slope and a Point
Make sure when you begin counting for a second point you begin at the given point, not at the origin.
Write an equation of a line by using its slope and any point on the line.
We can use the slope-intercept form to write the equation of a line if we know the slope and any point on the line.
The slope-intercept form is on the line.
CLASSROOM EXAMPLE 5
Using the Slope-Intercept Form to Write an Equation
Write an equation, in slope-intercept form, of the line having slope −2 and passing through the point (−1,4).
Slope on the line.
Given pointWrite an equation of a line by using its slope and any point on the line.
There is another form that can be used to write the equation of a line. To develop this form, let m represent the slope of a line and let (x1,y1) represent a given point on the line. Let (x, y) represent any other point on the line.
Definition of slope
Multiply each side by x − x1.
The point-slope form of the equation of a line with slope m passing through point (x1,y1) is
Write an equation of the line through (5,2), with the slope Give the final answer in slope-intercept form.
CLASSROOM EXAMPLE 6
Using the Point-Slope Form to Write Equations
Write an equation of a line by using two points on the line.
Many of the linear equations in Section 3.1−3.3 were given in the form
called standard form, where A,B, andC are real numbers and A and B are not both 0.
CLASSROOM EXAMPLE 7 line.
Writing the Equation of a Line by Using Two Points
Find an equation of the line through the points (2,5) and (−1,6). Give the final answer in slope-intercept form and standard form.
The same result would also be found by substituting the slope and either given point in slope-intercept form and then solving for b.
Write an equation of a line that fits a data set.
CLASSROOM EXAMPLE 8 line.
Writing an Equation of a Line That Describes Data
Use the points (3, 4645) and (7, 6185) to write an equation in slope-intercept form that approximates the data of the table. How well does this equation approximate the cost in 2005?
The equation gives y = 5415 when x = 5, which is a very good approximation.