This is an overview of the key ideas we have discussed during the first part of
this course.
You may find this summary useful as a study aid, but remember that the only way
to really
master the skills and understand the ideas is to practice solving problems .
The Addition Method (For Solving Systems of Linear Equations)
Write both equations in the form ax + by = c; multiply each equation by a
constant, if
necessary, so that the coefficients of one of the varibales are the same size;
then either add or
subtract the two equations in order to eliminate one of the variables; solve for
the remaining
variable ; then plug that value back into one of the original equations to solve
for the other
unknown .
Example: Solve the system of linear equations:
2x + 5y = 8
3x + 4y = 1
Solution: Begin by multiply the first equation by 3 and the second equation by
2, so that
the coefficients of x match up:
6x + 15y = 24
6x + 8y = 2
Then subtract the second eqution from the first to eliminate x :
7y = 22.
Hence
y =22/7.
Now we plug this into the original first equation to get
Therefore
Then we isolate x:
so
therefore
and consequently
So the answers are
Solving Absolute Value Equations
Example: Solve the equation 3x + 4 = 2.
Solution: Either
3x + 4 = 2 or 3x + 4 = −2
Hence
3x = −2 or 3x = −6
Therefore
The Point Slope Formula for a Line
Example: Find the equation for a line through the points (2, 5) and (−3, 2).
Solution: First we find the slope:
Now we plug this slope and either point into the formula y = y_{1} = m(x − x_{1})
to get
That is the equation for the line. If we need to, we can simplify and solve
for y:
so
Solving Quadratic Equations by Factoring
Example: Solve the equation x^{2} = 5x + 24 by factoring.
Solution: First, we need to move all the terms to one side of the equation, so
we subtract
5x from both sides, and we subtract 24 from both sides to get
x^{2} − 5x − 24 = 0
Now we factor the left side:
(x − 8)(x + 3) = 0
The only way this can be true is if one of the two factors is zero, so we see
taht either
x − 8 = 0 or x + 3 = 0
Consequently,
Solving Quadratic Equations by Completing the Square
Example: Solve the equation 3x^{2} = 10x + 6 by completing the square.
Solution: We need to get all the terms with x or x^{2} on the same side, so we
subtract 10x
from both sides:
3x^{2} − 10x = 6
We also need the coefficient of x to be 1, so we divide both sides of the
equation by 3:
Now we complete the square: the coefficient of x on the left side is
;
half of this is −5/3 ;the square of −5/3 is 25/9 , so that’s what we need to add
to both sides:
Since 2 = 18/9 , we can write this as
Now we factor the left side:
Then take square roots of both sides:
Therefore
Hence either
so
Function Notation
Example: Suppose that a function is given by the formula f(x) = x^{2} − 2x. Find
f(2) and
f(−3).
Solution: We evaluate the function by plugging the appropriate value in for x
and simpli
fying:
Similarly,
So we have
Example: Find a formula for a linear function f(x) that satisfies f(2) = 7
and f(4) = −1.
Solution: First we find an equation for a line that passes through the points
(2, 7) and
(4,−1) – we can so this using the pointslope formula:
Therefore
y − (−1) = −4(x − 4)
so
y + 1 = −4x + 16
Therefore
y = −4x + 15
Now we replace the y with f(x) to obtain function notation:
TI84 Calculator Reference Guide  Entering and Using Data
Tables
ENTERING LISTS
Most statistical calculations involve working with tables of data, so we need
to be able to input such tables into the graphing calculator . In the following
example, we will enter the data in the table at right. The same data will be
used in the later examples in this guide.
x 
2 
3 
7.2 
8.1 
108 
y 
5 
8 
9 
9 

1. Select the STAT menu by
pressing:
2. Select the first item in order to
edit the calculator’s lists:
3. Enter the xvalues in the first
column, labeled L1, as follows:
4. Use the right arrow key to move
the cursor to the first row of the
second column (labeled L2); then
enter the yvalues as follows:
SCATTERPLOTS
You can plot individual points on an xyplane by performing a scatterplot.
1. First, set the window as you
would when graphing a function.
To got to the menu for this,
press:
2. Go to the STAT PLOT menu by
pressing:
3. Select the first plot by pressing 1
or enter :
4. Select On by pressing enter:
5. View your plot by pressing:
REGRESSIONS
You can fit a line or curve to a set of data points by performing a regression
with the
graphing calculator. In this example, we find a line of best fit using a linear
regression.
1. Go to the STAT menu by
pressing:
2. Use the right arrow key to select
the CALC menu:
3. Select LinReg(ax+b) to perform a
linear regression on the data in
lists L1 (for xvalues) and L2(for
yvalues):
4. Calculate the coefficient for the
regression by pressing:
5. The resulting screen gives you
the coefficients for the
regression, as well as the
correlation coefficient r and its
square.