Practice Makes Perfect Algebra - Carolyn Wheater [10]
6. A(x, 6), B(6, 8), M(4, 7)
7. A(−1, 3), B(x, 9), M(3, 6)
8. A(−5, y), B(7, −3), M(1, 3)
9. A(4, −9), B(−2, y), M(1, −7)
10. A(0, 4), B(x, 0), M(8, 2)
* * *
Slope and rate of change
The slope of a line is a measurement of the rate at which it rises or falls. A rising line has a positive slope whereas a falling line has a negative slope as shown in Figure 4.2. The larger the absolute value of the slope, the steeper the line. A horizontal line has a slope of 0, and a vertical line has an undefined slope.
Figure 4.2 Lines with positive and negative slopes.
The slope of the line through the points(4, −1) and(0, 2) is
* * *
EXERCISE 4.4
Find the slope of the line that passes through the two points given.
1.(−5, 5) and(5, −1)
2.(6, −4) and(9, −6)
3.(3, 4) and(8, 4)
4.(4, 6) and(8, 7)
5.(7, 2) and(7, 5)
If a line has the given slope and passes through the given points, find the missing coordinate.
6. m = −4,(4, y) and(3, 2)
7. m = 2,(2, 9) and(x, 13)
10. m = 0,(−4, y) and(−6, 3)
* * *
Graphing linear equations
A linear equation in two variables has infinitely many solutions, each of which is an ordered pair(x, y). The graph of the linear equation is a picture of all the possible solutions.
Table of values
The most straightforward way to graph an equation is to choose several values for x, substitute each value into the equation, and calculate the corresponding values for y. This information can be organized into a table of values. Geometry tells us that two points determine a line, but when building a table of values, it is wise to include several more so that any errors in arithmetic will stand out as deviations from the pattern.
When you build a table of values, make a habit of choosing both positive and negative values for x. Of course, you can chose x = 0, too. Usually, you’ll want to keep the x-values near 0 so that the numbers you’re working with don’t get too large. If they do, you’ll need to extend your axes, or relabel your scales by 2s or 5s or whatever multiple is convenient. If the coefficient of x is a fraction, choose x-values that are divisible by the denominator of the fraction. This will minimize the number of fractional coordinates, which are hard to estimate.
* * *
EXERCISE 4.5
Construct a table of values and graph each equation.
1. y = 3x + 2
2. 2y = 4x − 8
3. 3y = 4x + 12
4. x + y = 10
5. y − 2x = 7
6. 6x + 2y = 12
7. 3x − 4y = 12
10. 4x − 3y = 3
Slope and y-intercept
To draw the graph of a linear equation quickly, put the equation in slope-intercept, or y = mx + b, form. The value of b is the y-intercept of the line, and the value of m is the slope of the line. Begin by plotting the y-intercept; then count the rise and run and plot another point. Repeat a few times and connect the points to form a line.
Intercept-intercept
If the linear equation is in standard, or ax + by = c, form, it is very easy to find the x- and y-intercepts of the line. The x-intercept is the point at which y equals 0, and the y-intercept is the point at which x equals 0. Substituting 0 for y reduces the equation to ax = c, and dividing by a gives the x-intercept. In the same way, substituting 0 for x gives by = c, and the y-intercept can be found by dividing by b. Plotting the x- and y-intercepts and connecting them will produce a quick graph.
* * *
EXERCISE 4.6
Use quick graphing techniques to draw the graph of each of the following equations.
2. 2x − 3y = 9
3. y = − 4x + 6
4. 6x + 2y = 18
5. x − 2y = 8
6. y = − 3x − 4
7. y − 6 = 3x + 1
8. 3x + 5y = 15
9. 2y = 5x − 6
10. 3x − 2y − 6 = 0
* * *
Vertical and horizontal lines
Horizontal lines fit the y = mx + b pattern, but since they have a slope of 0, they become y = b. Whatever value you may choose for x, the y-coordinate will be b.
Vertical lines have undefined slopes, so they cannot fit the y = mx + b pattern, but since every point on a vertical line has the same x-coordinate, they can be represented by an equation