Name __________________________________
Period __________
Date:
3-4
Finding an
Equation of a
Line
Topic:
Standard: A-REI.10
Essential Question: At times, architects and engineers must
derive an equation to describe a real-world problem. In
order to write a linear equation, what information would
they need?
Understand that the graph of an equation in two variables is the set of all
its solutions plotted in the coordinate plane, often forming a curve (which
could be a line).
Objective:
To find an equation of a line given its slope and a point on the
line, or two points, or its slope and the y-intercept.
Standard Form:
You know that the graph of
is a line in an
xy-plane. Equations of lines are often given in this standard
form, with A, B, and C integers. However, when you are asked
to find the equation of a certain line, the two other forms
discussed in this lesson may be more useful.
Point-Slope Form
By the theorem in the previous lesson, the line containing the
point (
) and having slope m has the equation
(
).
This equation is called the point-slope form of the equation of
the line.
Summary
Example 1:
A point and a slope Find an equation in standard form of the line containing the
given: point (4, 3) and having slope
.
Solution: Use the point-slope form with
(
)
(
(
Check:
Exercise 1:
( )
(
) and
)
(
)
(
(
.
)
)
)
Find an equation in standard form of the line containing the
point (2, 4) and having slope 0.4.
2
Example 2:
Two points given: Find an equation in standard form of the line containing the
points (1, 2) and (5, 1).
Solution: First, find the slope of the line:
(
)
Then use the point-slope form with either point.
(
(
Check:
)
)
(
(
)
(
(
)
)
)
3
Exercise 2:
Find an equation in standard form of the line containing the
points (3, 2) and (3, 2).
Complete this exercise twice: once with each point. Then,
check your answer using both points.
4
Example 3:
Slope and y-intercept Find an equation in standard form of the line having slope
given: and y-intercept .
The y-intercept is the point where the line crosses the y-axis.
Therefore the known point is ( ).
Solution: Use the point-slope form with
(
Exercise 3:
)
(
) and
(
)
Find an equation in standard form of the line having slope 1.2
and y-intercept 0.6.
5
Slope-intercept form: You could have used the slope intercept form of the linear equation to
solve the previous example and exercise.
If m is the slope and b is the y-intercept of a line, the equation
Is called the slope-intercept form of the equation of the line.
Nevertheless, the point-slope form of the linear equation will work
whenever you know the slope and the y-intercept. On the whole, the
point-slope form of the linear equation is more useful.
Exercise 3,
continued:
Find an equation in standard form of the line having slope 1 and
y-intercept 2.
Complete this exercise twice: once using the point-slope form and
once using the slope-intercept form of the linear equation.
6
You can explore the relationships between the slopes and the graphs
Parallel and
Perpendicular Lines: of two equations by graphing y = mx + b for various values of m and
b.
Graph
and
a different color pencil, graph
same grid.
on the grid below. Then using
and
on the
If you graph
and
for various values of m
and b on the same axes, your results should suggest the theorem
below.
This theorem tells how slopes indicate when graphs of equations are
parallel or perpendicular lines.
Let L1 and L2 be two different lines, with slopes m1 and m2
respectively.
1. L1 and L2 are parallel if and only if m1 = m2.
2.
L1 and L2 are perpendicular if and only if m1  m2 = 1.
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Parallel Lines:
The following lines are parallel,
,
because each has slope
.
Perpendicular Lines: The following lines are perpendicular,
because the product of the slopes is
(
Example 4:
)
.
Let P be the point (4, 1) and L the line
.
Parallel: Find an equation in standard form of the line L1 through P and
parallel to L.
Solution: Solve the equation of line L for y:

The slope of L is
.
To find an equation of L1, use
the point-slope form with
(
)
(
) and
:
8
Example 4
continued:
(
)
Therefore, the equation of the parallel line passing through the
given point is,
L1:
Perpendicular Find an equation in standard form of the line L2 through P and
perpendicular to L.
Solution: To find an equation of the perpendicular line L2, use
the pointslope form with the negative reciprocal slope: (
)
(
) and
:
(
)
Therefore, the equation of the perpendicular line passing
through the given point is,
L2:
9
Exercise 4:
Let P be the point (0, 1) and L the line
.
Find an equation in standard form of the line L1 through P and
parallel to L.
Then, find an equation in standard form of the line L2 through
P and perpendicular to L.
Class work:
p 120: 1-25
Homework:
p 121 Written Exercises: 2-30 even,
p 111 Written Exercises: 21-35 odd
p 121 Written Exercises: 31-55 odd
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