Writing equations given slope and point5.2

12/6/13

Solving Equations

• In any equation, we can solve for any variable that is not known

1) Rewrite in terms of that variable2) Find a value if all the other values are known3) Can use this to find equations

Point-Slope Form of the Equation of a Line

• The point-slope equation of a non-vertical line of slope m that passes through the point (x1, y1) is

y – y1 = m(x – x1).

Solving With Slope and a Point

1. Substitute known values into the equation

2. Rewrite the equation y = mx + b with slope and y-intercept values

Example: Writing the Point-Slope Equation of a Line

Write the point-slope form of the equation of the line passing through (-1,3) with a slope of 4. Then solve the equation for y.

Solution We use the point-slope equation of a line with m = 4, x1= -1, and

y1 = 3.

This is the point-slope form of the equation.y – y1 = m(x – x1)

Substitute the given values. Simply.y – 3 = 4[x – (-1)]

We now have the point-slope form of the equation for the given line.

y – 3 = 4(x + 1)

We can solve the equation for y by applying the distributive property.

y – 3 = 4x + 4

y = 4x + 7 Add 3 to both sides.

• Find the equation of the line passing through the point (-3, 0 ) and has slope of 1/3

Solving with Slope and Point

Solving with Slope and a Point

• Find the equation of the line passing through point (-2, -1) with slope of -3

Solving with Slope and Point

• Find the equation of the line passing through the point (-3, 0 ) and has slope of 1/3

Solving with Slope and a Point

• Find the equation of the line passing through (3 , -4) and is parallel to the line y = -3x – 2

Slope and Perpendicular Lines

90°

Slope and Perpendicular Lines• If two non-vertical lines are perpendicular, then

the product of their slopes is –1.• If the product of the slopes of two lines is –1,

then the lines are perpendicular.• A horizontal line having zero slope is

perpendicular to a vertical line having undefined slope.

Slope and Perpendicular Lines• If two non-vertical lines are perpendicular, then

the product of their slopes is –1.• If the product of the slopes of two lines is –1,

then the lines are perpendicular.• A horizontal line having zero slope is

perpendicular to a vertical line having undefined slope.

Two lines that intersect at a right angle (90°) are said to be perpendicular. There is a relationship between the slopes of perpendicular lines.

Perpendicular Lines

Perpendicular lines have negative reciprocal slopes so if you need the slope of a line perpendicular to a given line, simply find the slope of the given line, take its reciprocal (flip it over) and make it negative.

31

3m

3

1m

Example: Finding the Slope of a Line Perpendicular to a Given Line

Find the slope of any line that is perpendicular to the line whose equation is x + 4y – 8 = 0.

Solution We begin by writing the equation of the given line in slope-intercept form. Solve for y.

x + 4y – 8 = 0 This is the given equation.

4y = -x + 8 To isolate the y-term, subtract x and add 8 on both sides.

Slope is –1/4.

y = -1/4x + 2

Divide both sides by 4.

The given line has slope –1/4. Any line perpendicular to this line has a slope that is the negative reciprocal, 4.

Example: Writing the Equation of a Line Perpendicular to a Given Line

Write the equation of the line perpendicular to x + 4y – 8 = 0 that passes thru the point (2,8) in standard form.

Solution: The given line has slope –1/4. Any line perpendicular to this line has a slope that is the negative reciprocal, 4.So now we need know the perpendicular slope and are given a point (2,8). Plug this into the point-slope form and rearrange into the standard form.

y – y1 = m(x – x1)

y1 = 8 m = 4 x1 = 2

y – 8 = 4[x – (2)]

y - 8 = 4x - 8

-4x + y = 0

4x – y = 0 Standard form

11 xxmyy

Let's look at a line and a point not on the line

(2, 4)

Let's find the equation of a line parallel to y = - x that passes through the point (2, 4)

y = - x

What is the slope of the first line, y = - x ?

This is in slope intercept form so y = mx + b which means the slope is –1.

1

-14

Distribute and then solve for y to leave in slope-intercept form.

6 xy

So we know the slope is –1 and it passes through (2, 4). Having the point and the slope, we can use the point-slope formula to find the equation of the line

2

What if we wanted perpendicular instead of parallel?

(2, 4)

Let's find the equation of a line perpendicular to y = - x that passes through the point (2, 4)

y = - xThe slope of the first line is still –1.

11 xxmyy 1 24

Distribute and then solve for y to leave in slope-intercept form.

2xy

The slope of a line perpendicular is the negative reciporical so take –1 and "flip" it over and make it negative.

1

1

1

1

So the slope of a perpendicular line is 1 and it passes through (2, 4).

Homework:

• 5.2 worksheet #’s 1-15• Due Monday, 12/9