Prentice Hall's Online Lesson Quiz Enter Web Code: aga–0202
Students are often expected to "discover a basic truth" in understanding algebra on their own. This "truth," so to speak is how our alphabet is used in the language of algebra. Mathematicians have come to understand that the end of the alphabet, x,y, and z is used for variables (unknowns) while the beginning of the alphabet, a,b, and c is used to represent a constant (number value). Then, to add to our confusion, these are sometimes capitalized as A, B, and C.
In the last section, we discovered function notation, f(x), uses the middle of the alphabet f, g, and h when naming functions. Again, you might see functions named as F(x), G(x), etc.
Finally, there are letters that are "designated as
special." Mathematicians use m to represent the slope of a
line; e is used for the base of natural logarithms; and i
represents the imaginary unit of a complex number, 
.
Now you begin to understand the confusion that arises.
Standard form is merely a tool to "beautify equations." Notice that the x and y terms are contained on the left-hand side of the equation while the constant term is isolated to the right.
Also notice the constants are not fractions, the leading coefficient on the x-term is generally positive and it is possible for either A or B, but not both, to be zero.
The standard form of a linear equation is
Ax + By = C
where A, B, and C are real numbers and A and B are not both zero.
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Equations whose graphs are straight lines are called linear equations.
An equation can be identified as linear if:
 | Variables occur to the first power only. |
| There are no products of variables. |
| No variable appears in the denominator. |
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You can use set notation to write the solutions of the equation as {(x, y)|y = 2x + 1}. Read the notation as "the set of ordered pairs x, y such that y = 23x + 1." Since the values of y depend on the value of x, y is called the dependent variable and x is called the independent variable.
Method #1 – TABLE OF VALUES
A function can be graphed by making a table of values, graphing enough ordered pairs to see a pattern, and connecting the points with a straight line or smooth curve.
Example 2:
Graph: y = x – 3
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When asked to graph a function in standard form, one possible method of graphing is to locate the points where the graph intersects each axis and connect them with a line. In order for a line to cross an axis, one coordinate must be zero. As you can see in the table of values below, this is a rapid approach when values are integers.
Example 3: |
Graph: 3x + 2y = 6 |
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When x = 0, then y = 3; and, when y = 0, then x = 2. |
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Any two distinct points determine a line. The slope of the line is the ratio of the change in y-values to the change in x-values. The slope of a line is constant, that is, unchanging all along the line.
When calculating the slope between two
points, it is important to know that once you designate your points as
you must stay consistent when substituting values into the formula.
Example 4: |
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Find the slope between points P (– 3, 5) and Q (4, – 2)
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Another important observation are the two extreme conditions where zero might occur.
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If zero appears in the numerator, then you have a
graph of a horizontal line such as, y = a constant. |
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If zero appears in the denominator, then the graph
is that of a vertical line and the equation is x = a
constant. This condition is called undefined or no slope.
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It is vital that you understand the difference between an undefined (no slope) and a zero slope. |
Example 5: |
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Vertical lines do not have a slope! |
Horizontal lines have 0 for a slope. |
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x = 2 |
y = – 3 |
The purpose of this lesson is to learn to write equations for lines based on limited information. For example, you know a point and the y-intercept, or you know a point and a slope, or you know two points.
The difficulty for students does not seem to be in "how to" as much as, "when to" use the information they have been given.
You know a slope and the y-intercept.
You know a point and a slope, but not the y-intercept.
You know only two points on the line.
Since the first case is rather trivial, and reserved for beginning algebra students, it becomes important to know:
YOU MUST HAVE, AT LEAST, A POINT AND A SLOPE, OR TWO POINTS TO WRITE THE EQUATION OF ANY LINE.
Once students are thoroughly convinced of this fact, writing equations of lines becomes easier.
The point–slope form of an equation of a line is: y – y1 = m(x – x1). Use this form when you are given a point and a slope.
| Example 6: | ||
| Write in standard form the equation for a line
with slope of – 2 through the point ( –3, 1). |
y – 1 = –2(x
+ 3) y – 1 = –2x –6 y = –2x – 5 2x + y = – 5 |
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The slope–intercept form of an equation of a line is: y = mx +b. The slope is m and the y–intercept is b. Use this form when you are given a slope and the y-intercept.
| Example 7: | ||
| Write an equation for a line given the slope is 3 and y-intercept is 2. |
y = 3x + 2 |
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It is possible to find the slope of a line by examining the equation.
| Example 8: | ||
| Find the slope of 5x –
2y = 6
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The slope of the
line is
.
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Writing an Equation of a Perpendicular Line
Write an equation of the line that passes through the point (–2, 4) and is perpendicular to the line 2x + y = 8.
Do you understand that this problem is talking about two different lines but the two lines must be perpendicular to each other ? If so, you are on the right track.
Notice the given line has the equation 2x
+ y = 8.
By solving this equation for y = mx + b we should see the slope.
So, the slope of
the first line is m = – 2, but a line that runs perpendicular to this
line must have a slope of
.
Use the facts that x = – 2, y = 4, and in the
slope-intercept formula.
Therefore,
the slope of the first line is m = – 2, so the line that runs
perpendicular to it and contains the point (– 2, 4) has a slope of and
a y-intercept at 5.
Therefore,
the equation
of the second line is
, while the equation of
the first line is still y = – 2x + 8.
