Download Review of Linear Equations and Linear Functions

Review of Linear Equations and Linear Functions
A linear equation in one or two variables is an equation whose graph is a straight line.
There are many forms in which a linear equation may be written. For example, these are
all linear equations:
a)
b)
c)
d)
e)
f)
g)
h)
i)
x+y=1
2x = 1 + y
3y = x – 4
2y + 3x = 6
6 = 3x + 2y
x=5
-4 = y
x = -3x + 4
y = 2x + 3
There are two forms that are used more than the others. They are:
The Standard Form:
ax + by = c
For example:
2x + 3y = 6
And the Slope-Intercept Form:
For example
y = mx + b
y = 3x – 2
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To sketch the graph of a linear equation (which is a straight line), first draw a nice, neat
cartesian coordinate axis pair. This means using a ruler or straight edge to draw x- and yaxes that are perpendicular to each other and drawing an even, consistent scale on the
axes (evenly spacing the ticks for the numbers), and maybe even labeling the axes.
Good coordinate axes
Bad coordinate axes
Even better, you can use pre-printed graph paper. Google “free graph paper” and you will
find lots of web sites where you can print graph paper.
Next, find some points on the line. Make a table of values. Try to find the points where one
or the other variable is 0. These points are called the “intercepts”. They are the points
where the line crosses the axes.
So start with a table like:
x
0
y
0
Fill in the table and plot these points. If the points are very close to each other you should
add more points to your table. Use your ruler or straight edge to draw a line through the
points.
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Example: Graph the equation -3x + y = 6
Solution: make a table for the values of x and y, and start by filling in the value 0 for x in one
row, and the value 0 for y in another row:
x
y
0
0
Now calculate the value of y when x is 0, and the value of x when y is 0, and put these values
in the table. Try to do these calculations in your head, but here they are written down:
When x = 0, the equation becomes:
-3*0 + y = 6
y=6
And when y = 0, the equation becomes
-3x + 0 = 6
-3x = 6
Divide both sides by -3 to solve for x
x = -2
Now put these values in the table:
x
0
-2
y
6
0
Plot these points on your coordinate axes and use a straight edge to connect the dots.
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Graph each equation.
1. x + y = -7
2. y = -2x - 4
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Now that you have graphed some linear equations, let’s explore an important relationship.
Imagine moving from left to right along a line. Pick two points, close together, where the
line goes through grid points (i.e. points where the grid lines cross). To get from the first to
the second, count how many boxes over you have to go, and how many boxes up. Now do
the same thing for two points farther away from each other. And again for another pair of
points on the line. What do you notice?
The line in our first example goes though the
points (-2,0) and (-1,3).
To get from (-2,0) to (-1,3) you have to move
1 box over and 3 boxes up.
This line also goes through (-5,-9) and (1,9).
To get from (-5,-9) to (1,9) you have to move
6 boxes over and 18 boxes up.
The line also goes through (-3,-3) and (-1,3).
To get from (-3,-3) to (-1,3) you have to
move 2 boxes over and 6 boxes up.
What can you say about the ratio of “up boxes” to “over boxes”?
It is the same, 3, for every pair of points. This is true for all lines.
This ratio is sometimes referred to as “the rise over the run” and it is called the slope of
the line.
For each line that you have graphed:
Find the slope of the line by picking two points on the line, counting the “up boxes” and
“over boxes” to get from one to the other, and calculating the rise over the run. Note that if
the line slopes downward the number of “up boxes” will be negative.
Now put the equation of the line into the Slope-Intercept form, y = mx + b, by solving the
equation for y and find the value of m.
Compare the value of m with the “rise over the run” ratio.
So if you put the equation of a line in to Slope-Intercept form, y = mx + b, you can easily plot
the y-intercept, and then you can use the slope to find other points on the line and connect
the dots to graph the line.
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Example: Given the equation x + 2y = -8, put the equations into slope-intercept form. (i.e.
solve the equation for y) and graph the line.
Solution:
Slope-intercept form is y = mx + b
x + 2y = 8
-x + x + 2y = -x + 8
0 + 2y = -x + 8
2y = -x + 8
y = -1/2 x + 4
So:
1. the y-intercept is 4, which means that when x is 0 then y is 4, so the line goes
through the point (0,4)
2. the slope is -1/2, which means that if you go over two boxes you go down 1. If you
start at the point (0,4), the next point would be (2,3), and the one after that would
be (4,2)
Plot some points
Connect the dots
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More about the slope
If the rise is equal to the run, which means that the number
of “up boxes” is equal to the number of “over boxes” then
the ratio of “the rise over the run” is equal to 1.
This line slopes up at an angle of 45 degrees and its slope is
1
If the rise is greater than the run, the number of “up boxes”
is more than the number of “over boxes” and then the ratio
of “the rise over the run” is greater than 1.
This line slopes up at an angle of more than 45 degrees.
For every “over box” there are two “up boxes” so the rise
over the run is 2/1.
The slope of this line is 2
If the rise is less than the run, the number of “up boxes” is
less than the number of “over boxes” and then the ratio of
“the rise over the run” is a fraction, less than 1.
This line slopes up at an angle of less than 45 degrees. For
every 2 “over box” there I only 1 “up box” so the rise over
the run is 1/2.
The slope of this line is 1/2
If the line slopes downward, then the rise is negative. For a
positive number of “over boxes” there is a negative number
of “up boxes” so the ration of “the rise over the run” is
negative.
What is the slope of this line?
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Given two (or more) points on a line, find an equation of the line.
We will find the Slope-Intercept form of the equation, y = mx + b
Here are the steps:
1. Plot the points
2. Find the slope and substitute it for “m” in the y=mx+b equation
3. Choose a point, substitute the coordinates for x and y in the equation and solve for b
Example: Find the equation of the line through the points (1,-2), (4,4) and (6,8)
1. Plot the points
2. Find the slope: To get from the point
(4,4) to the point (6,8), you have to
go over 2 and up 4, so the rise over
the run is 4/2, which is 2, so the
slope is 2. Substitute this for m and
the equation becomes y = 2x + b
3. Choose the point (6,8), substitute 6
for x and 8 for y to get
8 = 2*6 + b
8 = 12+b
8-12 = 12-12+b
-4 = b
So the equation is y = 2x - 4
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Find an equation for each of these lines.
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Parallel Lines
Two lines are parallel to each other if they do not cross. If you start at a point with the
same x-coordinate on each line and move the same number of boxes over, then you must
also move the same number of boxes up or the lines would cross. So parallel lines have
the same slope.
For each of the lines that you graphed, draw line through the point (2,5) that is parallel to
it. Just start at the point (2,5) and use the ratio of “up boxes” to “over boxes” to find other
points on the line and “connect the dots” using your straight edge.
What is the y-intercept for each of these lines?
Write an equation for each of these lines.
Example: In the first example we graphed the equation -3x + y = 6 and here is our graph
with the point (2,5) added
We already figured out that the slope of our original
line is 3. We want to draw a parallel line, which
must also have slope 3, through the point (2,5).
If we start at the point (2,5) and go over 1, we must
go up 3, which would bring us to (3,8)
Going backwards, if we started at (2,5) and went
back 1, we must go down 3, which would bring us to
(1,2).
If we go back 1 more, we would go down 3 more to
the point (0,-1).
Let’s plot these points.
Now use a straight edge to draw a line through the
new points:
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The y-intercept of the new line is -1, so the equation is y = 3x - 1
Now let’s find the equations using algebra. Here are the steps:
1. Find the slope m of the original line.
2. Substitute 2 for x, 5 for y, and the slope of the original line for m in the equation
y = mx + b
3. Solve this equation for b
4. The equation of the parallel line is y = mx+ b where m is the slope of the original line
and b is the value you found in step 2.
Example #4.
Find the equation of the line parallel to y = 2x -4 passing through the point (2,5)
The equation of the original line is y = 2x – 4
1. The slope m of the original line is 2
2. Substituting 2 for x, 5 for y and 2 for
m in y = mx + b we get 5 = 2*2 + b
3. Solve for b:
5=4+b
5–4=4–4+b
1=0+b
1=b
4. The equation of the parallel line is
y = 2x + 1
Fact to remember: Lines are parallel if and only if they have the same slope.
Exception: A vertical line has no slope, and all vertical lines are parallel to each other.
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Suppose the equation of the line l1 is
Find the equation of another line l2 that is
parallel to l1 and passes through the point (6,3)
Suppose the equation of the line l1 is
Find the equation of another line l2 that
is parallel to l1 and passes through the point (5,-9)
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Linear Functions
A function is a rule, or formula, for calculating an output value from an input value.
The point-slope form of the equation of a line, y = mx + b, is a formula for calculating the
output value y from the input value x.
A function defined using this kind of formula is a linear function.
In function notation, the output value is often denoted as “f(x)”, so a linear function will
often be written in the form:
f(x) = mx + b
The graph of the function f(x) = mx + b is just the graph of the line y = mx+b
Example: Graph the linear function f(x) = 3x+6
Solution: The graph of this function is just the graph of the line y = 3x + 6
Make a table:
x y
0
0
Substitute 0 for x and solve for y:
y = 3*0 +6
y=6
Substitute 0 for y and solve for x
0=3x+6
-6=3x
-2=x
Fill in the table with these values
x
0
-2
y
6
0
Plot the points and connect them
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