Slopes and Equations of Lines

 Fact You can use this fact when you know: Slope formula: two points on a line Slope-intercept formula: y = mx + b the slope and y-intercept of a line Point-slope formula: y - y1 = m(x - x1) the slope of a line and a point on the line Parallel lines have equal slopes the slope of a line The slopes of perpendicular lines are opposite reciprocals the slope of a line

The most difficult part of working with points, slopes and lines is determining which formula to use when solving specific problems. When you are trying to solve a problem, ask yourself these questions:

1. What am I supposed to find?
2. What do I already know?
3. What method will I use?

The chart below outlines the methods that are appropriate to use when solving specific types of problems.

 What do you want to find? What do you know already? Method to use Slope of a line Coordinates of two points on the line Use the slope formula Slope and y-intercept of a line Equation of a line in standard form Write the equation in slope-intercept form Equation of a line Slope of the line and a point on that line Use the point-slope formula Equation of a line Slope and y-intercept of a line Use the slope-intercept formula Equation of a line Coordinates of two points on the line Use the slope formula to find the slope of the line, then use the slope and one of the points in the point-slope formula Equation of a line parallel to a given line Equation of the given parallel line and a point on your line Write the equation of the given line in slope-intercept form to determine its slope, then use that same slope and your point in the point-slope formula Equation of a line perpendicular to a given line Equation of the given perpendicular line and a point on your line Write the equation of the given line in slope-intercept form to determine its slope, then use the opposite reciprocal of that slope and your point in the point-slope formula

Let's look at some examples.

Example 1 What is the slope of a line through the points (2,3) and (4,-5) ?

What do we want to find? The slope of a line

What do we already know? Two points on the line

What method will we use? Use the slope formula

Example 2 What are the slope and y-intercept of 2x - 3y = 5 ?

What do we want to find? The slope and y-intercept

What do we already know? The equation of a line in standard form

What method will we use? Write the equation in slope-intercept form

2x - 3y = 5

- 3y = - 2x + 5

y = 2/3 x - 5/3

Since the equation now is in y = mx + b form, we can identify the slope m=2/3 and the y-intercept b=-5/3

Example 3 What is the equation of a line with slope - 2 which passes through (3,-5)?

What do we want to find? The equation of a line

What do we already know? The slope and a point on the line

What method will we use? Use the point-slope formula

m = -2, (x1, y1) = (3, -5)

y - y1 = m(x - x1)

y - -5 = -2(x - 3)

y + 5 = -2x + 6

y = -2x + 1

Example 4 What is the equation of a line with slope 3/4 and y-intercept - 3?

What do we want to find? The equation of a line

What do we already know? The slope and y-intercept

What method will we use? Use the slope-intercept formula

m = 3/4, b = -3

y = mx + b

y = 3/4 x + -3

y = 3/4 x - 3

Example 5 What is the equation of a line that contains the points (3,4) and (1,-2)?

What do we want to find? The equation of a line

What do we already know? Two points on the line

What method will we use? Use the slope formula to find the slope of the line, then use the slope and one of the points in the point-slope formula

Now use one of the points and our slope in the point-slope formula:

m = 3

y - y1 = m(x - x1)

y - 4 = 3(x - 3)

y - 4 = 3x - 9

y = 3x - 5

Example 6 Find the equation of a line that goes through (-1, 4) and is parallel to 3x-y=5. What do we want to find? The equation of a line

What do we already know? A point on our line and the equation of a line parallel to our line

What method will we use? Write the equation of the given line in slope-intercept form to determine its slope, then use that same slope and your point in the point-slope formula

3x - y = 5

- y = - 3x + 5

y = 3x - 5

m = 3, b = - 5

So the slope of the given line is 3

Parallel lines have equal slopes, so the slope of our line is also 3. Our line also goes through the point (-1,4), so we'll write an equation with the point-slope formula.

y - y1 = m(x - x1)

y - 4 = 3(x - - 1)

y - 4 = 3(x + 1)

y - 4 = 3x + 3

y = 3x + 7

Example 7 Find the equation of the line through (4,-1) that is perpendicular to 4x+3y=2

What do we want to find? The equation of a line

What do we already know? A point on our line and the equation of a line perpendicular to our line

What method will we use? Write the equation of the given line in slope-intercept form to determine its slope, then use the opposite reciprocal of that slope and your point in the point-slope formula

4x + 3y = 2

3y = - 4x + 2

y = - 4/3 x + 2/3

m = - 4/3, b = 2/3

Perpendicular lines have slopes that are opposite reciprocals, so the slope of our line is the opposite reciprocal of -4/3, or 3/4. Our line also goes through the point (4,-1), so we'll write an equation with the point-slope formula.

y - y1 = m(x - x1)

y - - 1 = 3/4(x - 4)

y + 1 = 3/4 x - 3

y = 3/4 x - 4