More Slopes

In Chapter 6, you investigated the slope-intercept and point-slope equations. These equations allowed you to become familiar with a number of activities you can perform to explore relationships between sets of numbers. You can view such equations as functions. As explained previously, you can view a function as a formal mechanism for interpreting or transforming the values of a domain into those of a range.

Making It Easy

When you create a linear equation, you can make use of the point-slope equation. As the discussion in Chapter 6 indicates, expressed in its entirety, this equation takes this form:

(y − y₁) = m (x − x₁)

This equation is for a line with slope m that contains the point (x₁, y₁). The slope-intercept equation also uses a slope and a point. The point is called the y-intercept. The equation appears in one of two forms:

y = mx + b

Ax + By = C

Drawing on the discussion in Chapter 6, consider a situation in which you know the slope of a line is 2. You can then write the following, preliminary equation of a line:

y = 2x + b

If you know the coordinates of a point on the line, then you can substitute the x and y values that define the point into the slope-intercept equation. Assume, for example, that you are working with the point (4, 11). You can substitute the x and y values of this point into the standard slope-intercept equation in this way:

11 = 2(4) + b

Having made this substitution, you can then solve for b:

11 = 8 + b

11 − 8 = b

3 = b

Having solved the equation for b, you can then substitute the values of the slope and b (the y-intercept) into the original slope-intercept equation to create the equation for your line:

y = 2x + 3

Making It Still Easier

In Chapter 6, you also dealt with the ratio that exists between the rise and run of a line. You expressed that ratio in this way:

The Greek letter delta signifies changes in the ratio. The ratio works for all ordered pairs on a line. If you return to the ordered pair you dealt with in the previous section (4, 11), you can substitute as follows:

Since you know the slope of the function is 2, you can set up this equation:

You can then solve the equation for y:

2(x − 4) = y − 11

2x − 8 = y − 11

2x − 8 + 11 = y

2x + 3 = y

Again, given the basic versions of the point-slope form of a linear function,

m (x₂ − x₁) = (y₂ − y₁)

or

(y₂ − y₁) = m (x₂ − x₁),

you can then proceed to develop different equations with relative ease. You derive the slope using a ratio of the values you find in two points anywhere on the line:

You then proceed to take any ordered pair on the line and substitute it into a version of the point-slope equation. If you are working with a line that has a slope of −3, for example, and you know that the ordered pair (2, −7) lies on the line, then you substitute in this way:

y − (−7) = −3(x − 2)

or

y + 7 = −3(x − 2)

You then solve these for y to arrive at the slope-intercept form:

y + 7 = −3x − 3(−2)

y + 7 = −3x + 6

y = −3x − 1

Exercise Set 7.1 Write the slope-intercept equation for the line containing the given pair of points. (0, 0) and (12, 4) (0, 2) and (12, 8) (0, 4) and (10, 12) (0, 0) and (2, 10) (0, 2) and (12, 6) Write the point-slope equation for the line passing through each pair of points. (−2, 7) and (4, −3) (1, 2) and (3, 7) (−3, 1) and (4, 3) (2, 1) and (7, 2) (−1, 2) and (3, 4)