Transcript Chapter 7 Power Point

Solving Equations
and Inequalities
Chapter 7
Solving Two-Step
Equations
Section 7-1
How to Solve a
Two-Step Equation
• One goal in solving an equation is to have
only variables on one side of the equal
sign and numbers on the other side of the
equal sign.
• The other goal is to have the number in
front of the variable equal to one.
– The variable does not always have to be x.
These equations can make use of any letter
as a variable.
How to Solve Two-Step
Equations continued…
The most important thing to remember in
solving a linear equation is that whatever
you do to one side of the equation, you
MUST do to the other side.
– So if you subtract a number from one side,
you MUST subtract the same value from the
other side. You will see how this works in the
examples.
Solving Two-Step Equations
continued…
Solving a two-step equation requires the
same procedure(s) as a one-step equation.
However, the order in which the procedures
are done makes a difference.
Do the inverse operation for
addition or subtraction first.
Do the inverse operation of
multiplication or division last.
Inverse
The
operation
that reverses
the effect of
another
operation.
Solving Multi-Step
Equations
Section 7-2
Solving the Equation
Solve 3x – 9 = 33
Step 1: You need to get the first term
with the variable by itself. So you need to
“UNDO” or get rid of -9. To do this, you
do the OPPOSITE of what is being done.
You are currently SUBTRACTING 9; the
OPPOSITE is ADDITION. What you do
to one side, you MUST do to the other.
So you will ADD 9 to both sides.
Step 2: The variable is still not by itself,
so you need to do the OPPOSITE of what
is being done. 3 is being multiplied to X
and the OPPOSITE of multiply is DIVIDE.
So you DIVIDE both sides by 3.
The X is finally alone which means you
have completed the equation.
Finding Consecutive
Integers
• The definition of consecutive is “following one
another in uninterrupted intervals.”
• When you count by 1’s from any integer, you are
counting consecutive integers.
Four consecutive integers
Three consecutive integers
1, 2, 3, 4
The sum of three consecutive integers
Let n = the least integer.
Then n + 1 = the second integer,
and n + 2 = the third integer.
n+n+1+n+2
=
-5, -4, -3
is
96
96
Steps for Solving a
Multi-Step Equation
Step 1: Use the Distributive Property, if
necessary.
Step 2: Combine like terms.
Step 3: Undo addition or subtraction.
Step 4: Undo multiplication or division.
Let’s Practice
Multi-Step Equations with
Fractions and Decimals
Section 7-3
Solving Multi-Step Equations
with Fractions
To clear a fraction from an
equation, you multiply
both sides of the equation
by the denominator.
4x=12
5
Examples
2n-6=22
3
-7k+14=-21
10
1x+3=2
4
How to Clear Equations
of Fractions
TO SOLVE AN EQUATION WITH fractions, we
transform we transform it into an equation without
fractions – which we know how to solve. The
technique is called clearing of fractions.
Example 1
Solve for x:
x+x−2=6
3
5
Caution! Be sure the
distributive law is used to
multiply all of the terms by 15.
Multiply both sides of the equation--every term--by
the LCM of denominators. Every denominator will
then cancel. We will then have an equation without
fractions. The LCM of 3 and 5 is 15. Therefore,
multiply every term on both sides of the equal sign
by 15. Each denominator will now cancel into 15-that is the point--and we have the following simple
equation that has been "cleared" of fractions.
How to Clear Equations
of Decimals
To clear an equation of decimals, we count the
greatest number of decimal places in any one
number. If the greatest number of decimal places
is 1, we multiply both sides by 10; if it is 2, we
multiply by 100; and so on.
Example 2
Solve: 16.3 - 7.2y = -8.18
Solution
The greatest number of decimal
places in any one number is two.
Multiplying by 100 will clear all
decimals.
Quick Check
Solve each equation:
a. -7 + y = 1
12
6
b. 1b – 1 = 5
3
6
c. 1.5x – 3.6 = 2.4
d. 1.06p – 3 = 0.71
Write an Equation
Section 7-4
Steps to Writing Equations
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Read the problem carefully and figure out what it is asking you to find.
– Usually, but not always, you can find this information at the end of the problem.
Assign a variable to the quantity you are trying to find.
– Most people choose to use x, but feel free to use any variable you like.
Write down what the variable represents.
– By the time you read the problem several more times and solve the equation, it
is easy to forget where you started.
Re-read the problem and write an equation for the quantities given in the
problem.
– The only way to truly master this step is through lots of practice. Be prepared to
do a lot of problems.
Solve the equation.
Answer the question in the problem.
– Just because you found an answer to your equation does not necessarily mean
you are finished with the problem. Many times you will need to take the answer
you get from the equation and use it in some other way to answer the question
originally given in the problem.
Check your solution.
– Your answer should not only make sense logically, but it should also make the
equation true.
Solving Equations with
Variables on Both Sides
Section 7-5
How to Solve an Equation with
Variables on Both Sides
• Consider the equation x – 6 = –2x + 3. To
isolate the variable, we need to get all the
variable terms to one side and the constant
terms to the other side.
– You may need to use the Distributive Property to
simplify one or both sides of an equation before you
can get the variable alone on one side.
• The first step is to use addition or subtraction to
collect the variable on one side of the equation.
• Next, we combine like terms and then isolate
the variable by multiplying or dividing.
Let’s Practice…
Example:
Solve
x – 6 = –2x + 3
Solution:
Step 3: Divide or multiply to isolate
the variable
3x = 9
Step 1: Get all the variable terms to
one side and the constant terms to
the other side.
x – 6 = –2x + 3
Check:
x – 6 = –2x + 3
Step 2: Combine like terms
2x + x = 3 + 6
3x = 9