Lines in the Coordinate Plane

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Lines in the Coordinate Plane
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Warm Up Substitute the given values of m, x, and
y into the equation y mx b and solve for
b. 1. m 2, x 3, and y 0 Solve each
equation for y. 3. y 6x 9
b 6
2. m 1, x 5, and y 4
b 1
y 6x 9
4. 4x 2y 8
y 2x 4
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Objectives
Graph lines and write their equations in
slope-intercept and point-slope form. Classify
lines as parallel, intersecting, or coinciding.
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Vocabulary
point-slope form slope-intercept form
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The equation of a line can be written in many
different forms. The point-slope and
slope-intercept forms of a line are equivalent.
Because the slope of a vertical line is
undefined, these forms cannot be used to write
the equation of a vertical line.
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Example 1A Writing Equations In Lines
Write the equation of each line in the given form.
the line with slope 6 through (3, 4) in
point-slope form
Point-slope form
y y1 m(x x1)
y (4) 6(x 3)
Substitute 6 for m, 3 for x1, and -4 for y1.
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Example 1B Writing Equations In Lines
Write the equation of each line in the given form.
the line through (1, 0) and (1, 2) in
slope-intercept form
Find the slope.
Slope-intercept form
y mx b
0 1(-1) b
Substitute 1 for m, -1 for x, and 0 for y.
1 b
Write in slope-intercept form using m 1 and b
1.
y x 1
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Example 1C Writing Equations In Lines
Write the equation of each line in the given form.
the line with the x-intercept 3 and y-intercept
5 in point slope form
Use the point (3,-5) to find the slope.
y y1 m(x x1)
Point-slope form
Simplify.
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Check It Out! Example 1a
Write the equation of each line in the given form.
the line with slope 0 through (4, 6) in
slope-intercept form
Point-slope form
y y1 m(x x1)
Substitute 0 for m, 4 for x1, and 6 for y1.
y 6 0(x 4)
y 6
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Check It Out! Example 1b
Write the equation of each line in the given form.
the line through (3, 2) and (1, 2) in
point-slope form
Find the slope.
y y1 m(x x1)
Point-slope form
Substitute 0 for m, 1 for x1, and 2 for y1.
y 2 0(x 1)
y - 2 0
Simplify.
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Example 2A Graphing Lines
Graph each line.
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Example 2B Graphing Lines
Graph each line.
y 3 2(x 4)
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Example 2C Graphing Lines
Graph each line.
y 3
The equation is given in the form of a horizontal
line with a y-intercept of 3. The equation
tells you that the y-coordinate of every point
on the line is 3. Draw the horizontal line
through (0, 3).
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Check It Out! Example 2a
Graph each line.
y 2x 3
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Check It Out! Example 2b
Graph each line.
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Check It Out! Example 2c
Graph each line.
y 4
The equation is given in the form of a horizontal
line with a y-intercept of 4. The equation
tells you that the y-coordinate of every point
on the line is 4. Draw the horizontal line
through (0, 4).
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A system of two linear equations in two variables
represents two lines. The lines can be parallel,
intersecting, or coinciding. Lines that coincide
are the same line, but the equations may be
written in different forms.
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Example 3A Classifying Pairs of Lines
Determine whether the lines are parallel,
intersect, or coincide.
y 3x 7, y 3x 4
The lines have different slopes, so they
intersect.
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Example 3B Classifying Pairs of Lines
Determine whether the lines are parallel,
intersect, or coincide.
Solve the second equation for y to find the
slope-intercept form.
6y 2x 12
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Example 3C Classifying Pairs of Lines
Determine whether the lines are parallel,
intersect, or coincide.
2y 4x 16, y 10 2(x - 1)
Solve both equations for y to find the
slope-intercept form.
2y 4x 16
y 10 2(x 1)
2y 4x 16
y 10 2x - 2
y 2x 8
y 2x 8
Both lines have a slope of 2 and a y-intercept of
8, so they coincide.
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Check It Out! Example 3
Determine whether the lines 3x 5y 2 and 3x
6 -5y are parallel, intersect, or coincide.
Solve both equations for y to find the
slope-intercept form.
3x 5y 2
3x 6 5y
5y 3x 2
Both lines have the same slopes but different
y-intercepts, so the lines are parallel.
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Example 4 Problem-Solving Application
Erica is trying to decide between two car rental
plans. For how many miles will the plans cost the
same?
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The answer is the number of miles for which the
costs of the two plans would be the same. Plan A
costs 100.00 for the initial fee and 0.35 per
mile. Plan B costs 85.00 for the initial fee and
0.50 per mile.
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Write an equation for each plan, and then graph
the equations. The solution is the intersection
of the two lines. Find the intersection by
solving the system of equations.
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Plan A y 0.35x 100
Plan B y 0.50x 85
Subtract the second equation from the first.
x 100
Solve for x.
Substitute 100 for x in the first equation.
y 0.50(100) 85 135
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The lines cross at (100, 135).
Both plans cost 135 for 100 miles.
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Check your answer for each plan in the original
problem. For 100 miles, Plan A costs 100.00
0.35(100) 100 35 135.00. Plan B
costs 85.00 0.50(100) 85 50 135, so
the plans cost the same.
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Check It Out! Example 4
What if? Suppose the rate for Plan B was also
35 per month. What would be true about the lines
that represent the cost of each plan?
The lines would be parallel.
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Lesson Quiz Part I
Write the equation of each line in the given
form. Then graph each line.
1. the line through (-1, 3) and (3, -5) in
slope-intercept form.
y 2x 1
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Lesson Quiz Part II
Determine whether the lines are parallel,
intersect, or coincide.
3. y 3 x,
y 5 2(x 3)
intersect
4. 2y 4x 12, 4x 2y 8
parallel