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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 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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**A line with y-intercept b contains the point (0, b).**

A line with x-intercept a contains the point (a, 0). Remember!

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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 y – y1 = m(x – x1) Point-slope form 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 Substitute for m, 3 for x1, and 0 for y1. 5 3 y – 0 = (x – 3) 5 3 y = (x - 3) 5 3 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 y – y1 = m(x – x1) Point-slope form Substitute 0 for m, 4 for x1, and 6 for y1. y – 6 = 0(x – 4) y = 6

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**Write the equation of each line in the given form.**

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. The equation is given in the slope-intercept form, with a slope of and a y-intercept of 1. Plot the point (0, 1) and then rise 1 and run 2 to find another point. Draw the line containing the points. (0, 1) rise 1 run 2

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**Example 2B: Graphing Lines**

Graph each line. y – 3 = –2(x + 4) The equation is given in the point-slope form, with a slope of through the point (–4, 3). Plot the point (–4, 3) and then rise –2 and run 1 to find another point. Draw the line containing the points. (–4, 3) rise –2 run 1

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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). (0, –3)

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**Check It Out! Example 2a Graph each line. y = 2x – 3**

The equation is given in the slope-intercept form, with a slope of and a y-intercept of –3. Plot the point (0, –3) and then rise 2 and run 1 to find another point. Draw the line containing the points. (0, –3) rise 2 run 1

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**Check It Out! Example 2b Graph each line.**

The equation is given in the point-slope form, with a slope of through the point (–2, 1). Plot the point (–2, 1)and then rise –2 and run 3 to find another point. Draw the line containing the points. (–2, 1) run 3 rise –2

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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). (0, –4)

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**A system of two linear equations in two variables represents two lines**

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 Both lines have a slope of , and the y-intercepts are different. So the lines are parallel.

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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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**Understand the Problem**

1 Understand the Problem The answer is the number of miles for which the costs of the two plans would be the same. Plan A costs $ 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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2 Make a Plan 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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Solve 3 Plan A: y = 0.35x + 100 Plan B: y = 0.50x + 85 Subtract the second equation from the first. 0 = –0.15x + 15 x = 100 Solve for x. Substitute 100 for x in the first equation. y = 0.50(100) + 85 = 135

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Solve Continued 3 The lines cross at (100, 135). Both plans cost $135 for 100 miles.

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Look Back 4 Check your answer for each plan in the original problem. For 100 miles, Plan A costs $ $0.35(100) = $100 + $35 = $ Plan B costs $ $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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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.

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.

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