Algebra Review!
Objectives:
1. To solve linear
equations, systems
of linear equations,
and quadratic
equations
Assignment:
• Algebra Review
Worksheet (Online)
Objective 1
You will be able solve linear equations, systems of
linear equations, and quadratic equations
Geometry and Algebra
For the most part, in Algebra, you were given
an equation that you had to solve, graph,
or perform some other kind of
mathematical wizardry.
In Geometry, where there are equations, you
generally have to set them up yourself.
Oh, then you have to solve them, of
course.
Linear Equations
A linear equation in one variable is one that
can be written in the form ax + b = 0.
This basically
means that there is
only one variable
and the highest
power is understood
to be one
A solution is
the number
that makes the
equation a
true statement
Solving Linear Equations
Solving linear
equations is super
simple. The goal
is to isolate the
variable.
Like Terms
Coefficients
7𝑥 2 + 5𝑥 + 1
Variables
Constant
To combine like terms,
just add/subtract the
coefficients
Like terms have the same variable raise to the
same power
Keep It Balanced
𝒙+𝟏=𝟐+𝟏
𝒙−𝟑=𝟐−𝟑
𝒙=𝟐
𝟒∙𝒙=𝟒∙𝟐
𝒙 𝟐
=
𝟓 𝟓
Inverse Operations
Operation
Inverse
Addition
Subtraction
Multiplication
Division
Squaring
Taking the
Square Root
Cubing
Taking the
Cube Root
Exponentiation
Taking the
Logarithm
Exercise 1
Solve for x.
4
 x  12
3
Exercise 2
Solve for x.
5
3 1
7
x  x
8
8 2
8
Protip #1
If your equation has a bunch of fractions,
make them mathemagically disappear by
multiplying each term by the common
denominator of all the fractions.
2
7 1
5
x  x
3
12 2
18
System of Equations
A system of equations is a collection of 2 or
more equations, linear or not, with the
same variables.
The solution to a
system of equations is
the set of all points (x, y)
that satisfy all the
equations in the system.
In general, for a
system to be
solvable, you need
one equation for
every variable in
the system.
Linear System of Equations
We have a linear system of equations
when the system consists of two variables,
x and y.
The solution to the system is the ordered pair
(x, y) that satisfies both equations.
Solving Linear Systems
There are a variety of methods to solve a linear system of
equations:
Graphing:
Graph the
1 see
lines and
where they
intersect
Substitution:
Substitute an
expression
containing
2 one
variable in the
place of another
variable
Elimination:
Add or subtract
a multiple of the
3
equations to
eliminate one
variable
Substitution Method
Only use this method if it is easy and
convenient to solve one of the equations
for one of the variables.
Solve one
of the
equations
for one of
the
variables
Substitute this
expression
into the other
equation and
solve for the
other variable
Substitute this
value into one
of the original
equations and
solve for the
other value
Exercise 3
Solve for x and y.
y  2x 1
2x  y  3
Protip #2
When solving a system of equations by
substitution, never substitute an
expression back into its own equation.
That’s like a dog chasing its own tail.
y  2x 1
2x  y  3
Elimination Method
Using the
Elimination
Method, you
are trying to get
rid of (eliminate)
one of the
variables by
adding or
subtracting the
equations.
8𝑥 + 2𝑦 = 4
−2𝑥 + 3𝑦 = 13
In the above system, let’s
say that you’re trying to
eliminate 𝑥. What you
would have to do first?
Elimination Method
Multiply 1 or
all of the
Step 1
equations
by a
constant
You want to make
a set of variables
differ only by sign
Add the new
equations,
Step 2
eliminating
one of the
variables
Substitute
this value
Step
into
one3 of
the original
equations
Solve for the
other variable
Exercise 4
Solve for x and y.
6 x  5 y  19
2x  3y  5
Quadratic Equations
The standard form of a quadratic equation in one
variable is ax2 + bx + c = 0, where a is not zero.
Highest power is 2,
that’s the quadratic
term
Quadratic equations
can have 0, 1, or 2
solutions
Solving Quadratic Equations
There are also a metric ton of ways to solve
quadratic equations:
1. Graphing: Graph the equation and see
where it intersects the x-axis
2. Factoring: Put the equation in standard
form, factor, and then apply the Zero
Product Property
 b  b 2  4ac
3. Quadratic Formula: x 
2a
Zero Product Property
If the product of A and B equals zero, what
must be true about A or B?
Zero Product Property
If the product of
two expressions is
zero, then at least
of the expressions
equal zero.
A B  0
Maybe that’s zero
Or maybe this one’s zero
(Or maybe they’re both zero)
Solving Quadratic Equations
The standard form of a quadratic equation
in one variable is ax2 + bx + c = 0, where a
is not zero.
We can use the zero product property to
solve certain quadratic equations in
standard form if we can write ax2 + bx + c
as a product of two expressions. To do
that, we have to factor!
Exercise 5
Solve 0 = x2 – 6x – 7.
0  x2  6x  7
0   x  7  x  1
Set each
factor equal
to zero
x7  0
x7
Factor
x 1  0
x  1
x-intercepts: (7, 0) and (−1, 0)
Solving Quadratic Equations
To solve a quadratic
equation, try applying
the zero product
property.
Factor
Step
your1
quadratic
Set each
factor
equal
Step to
2
zero and
solve
Exercise 6
Solve for x.
x  3 x  18
2
Exercise 7
Solve for x.
 4 x  12 x  7  0
2
The Quadratic Formula
Let a, b, and c be real numbers, with a ≠ 0.
The solutions to the quadratic equation
ax2 + bx + c = 0 are
b  b  4ac
x
2a
2
Song 1:
Song 2:
Exercise 8
Solve for x.
2x  7  x
2
Game
In this game, you will have to work
cooperatively in your groups to try to get
the most number of points. Certain
questions are worth between 1 and 5
points each. There are probably more
questions here that you have time to
complete, so strategize. We’ll stop 5
minutes before the bell to check answers
and tally scores.
Assignment
• Online Algebra
Review Worksheet
“Set it equal to zero!”