2.5 - 1 point-slope form the line with slope m passing through the point (x 1, y 1 ) has an equation...

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Point-Slope Form

The line with slope m passing through the point (x1, y1) has an equation

the point –slope form of the equation of a line.

1 1( ),y y m x x

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Example 1 USING THE POINT-SLOPE FORM (GIVEN A POINT AND THE SLOPE)

Find an equation of the line through (– 4, 1) having slope – 3. Solution

1 1Here 4, 1, and 3.x y m

1 1y y m x x Point-slope form

3 )1 4(y x 1 1 34, 1,x my

1 3 4y x Be careful with signs.

1 3 12y x Distributive property

3 11y x Add 1.

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Example 2 USING THE POINT-SLOPE FORM (GIVEN TWO POINTS)

Find an equation of the line through (– 3, 2) and (2, – 4). Solution Find the slope first.

( 3 5

4 62

2 )m

Definition of slope

1 1

6The slope is . Either ( 3,2) or (2, 4)

5can be used for , . We choose 3,2 .

m

x y

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Find an equation of the line through (– 3, 2) and (2, – 4). Solution

Point-slope form 1 1y y m x x

6

52 )3(y x x1 = – 3, y1 = 2, m = – 6/5

5 2 6 3y x Multiply by 5.

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Find an equation of the line through (– 3, 2) and (2, – 4). Solution

5 2 6 3y x Multiply by 5.

5 10 6 18y x Distributive property.

6 8

5 5y x Add 10; divide by 5.

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Slope-Intercept Form

As a special case,suppose that a line passes through the point (0, b), so the line has y-intercept b. If the line has slope m, then using the point-slope form with x1 = 0 and y1 = b gives

1 1y m xy x

0y xb m

y mx b

Slope y-intercept

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Slope-Intercept Form

The line with slope m and y-intercept b has an equation

the slope-intercept form of the equation of a line.

,y mx b

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Example 4 USING THE SLOPE-INTERCEPT FORM (GIVEN TWO POINTS)

Find an equation of a line through (1, 1) and (2,4). Then graph the line using the slope-intercept form.Solution Use the slope intercept form. First, find the slope.

23

4 1 3

1 1m Definition of slope.

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Example 4

Find an equation of a line through (1, 1) and (2,4). Then graph the line using the slope-intercept form.Solution

Slope-intercept form

Substitute 3 for m in y = mx + b and choose one of the given points, say (1, 1), to find b.

y mx b

1 3 1 b m = 3, x = 1, y = 1

2b Solve for b.y-intercept

USING THE SLOPE-INTERCEPT FORM (GIVEN TWO POINTS)

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Example 4

Solution The slope intercept form is 3 2y x

(2, 4)

(1, 2)

(0, – 2)y changes 3 unitsx changes 1 unit

USING THE SLOPE-INTERCEPT FORM (GIVEN TWO POINTS)

x

y

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Equations of Vertical and Horizontal lines

An equation of the vertical line through the point (a, b) is x = a.

An equation of the horizontal line through the point (a, b) is y = b.

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Parallel Lines

Two distinct nonvertical lines are parallel if and only if they have the same slope.

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Perpendicular Lines

Two lines neither of which is vertical, are perpendicular if and only if their slopes have a product of – 1. Thus, the slopes of perpendicular lines, neither of which are vertical, are negative reciprocals.

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Example 6 FINDING EQUATIONS OF PARALLEL AND PERPENDICULAR LINES

Find the equation in slope-intercept form of the line that passes through the point (3, 5) and satisfies the given condition.

Solution (3, 5) is on the line so we need to find the slope to use the point-slope form. Write the equation in the slope-intercept form (solve for y).

a. parallel to the line 2x + 5y = 4

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Solution


2 5 4x y 5 2 4y x Subtract 2x.

25

45

y x Divide by 5.

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Solution


25

45

y x

The slope is – 2/5. Since the lines are parallel, – 2/5 is also the slope of the line whose equation is to be found.

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Solution


1 1y y m x x Point-slope form

25

5 3y x m = – 2/5, x1 = 3, y1 = 5

2 65

5 5y x Distributive property

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Solution


2 65


2 315 5

y x Add 5 (25/5).

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Solution We know the slope of the line, so the slope of any line perpendicular to it is 5/2.

b. perpendicular to the line 2x + 5y = 4

1 1y y m x x

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Solution

b. perpendicular to the line 2x + 5y = 4

1 1y y m x x

52

5 3y x 1 13 5

5,

2,xm y

5 155


5 52 2

y x Add 5 (10/2).

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Equation Description When to Usey = mx +b Slope-Intercept Form

Slope is m.y-intercept is b.

Easily identified and used to quickly graph the equation.

y – y1 = m(x – x1)

Point-Slope FormSlope is m.Line passes through (x1, y1)

Ideal for finding the equation of a line if the slope and a point on the line or two points on the line are known.

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Equation Description When to UseAx + By = C Standard Form

(If the coefficients and constants are rational, then A, B, and C are expressed as relatively prime integers, with A≥ 0).

The x- and y- intercepts can be found quickly and used to graph the equation. The slope must be calculated.

Slope is ( 0).A

BB

-intercept is ( 0).C

x AA

-intercept is ( 0).C

y BB

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Equation Description When to Usey = b Horizontal Line

Slope is 0.y-intercept is b.

If the graph intersects only the y-axis, then y is the only variable in the equation.

x = a Vertical LineSlope is undefined.x-intercept is a.

If the graph intersects only the x-axis, then x is the only variable in the equation.

2.5 - 1 point-slope form the line with slope m passing through the point (x 1, y 1 ) has an equation...

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