Download Linear Equations in One Variable and Proportions

Ch. 6
Equations and Inequalities
6.2 Linear Equations in One Variable and
Proportions
Objectives
1.
2.
3.
4.
5.
Solve linear equations.
Solve linear equations containing fractions.
Solve proportions.
Solve problems using proportions.
Identify equations with no solution or infinitely
many solutions.
Linear Equations
 linear equation in one variable x
is an equation that can be written in the form
ax + b = 0,
where a and b are real numbers, and a  0.
 Solving an equation in x
determining all values of x that result in a true
statement when substituted into the equation.
 Equivalent equations
have the same solution set.
4x + 12 = 0 and x = 3 are equivalent equations.
Addition and Multiplying Properties
 The Addition Property of Equality
If a = b, then a + c = b + c
 The Multiplication Property of Equality
If a = b, then a · c = b · c
Using Properties of Equality to Solve
Equations
Solving Linear Equation
Solve and check: 2(x – 4) – 5x = 5.
Step 1. Simplify the algebraic expression on each side:
2(x – 4) – 5x = 5
2x – 8 – 5x = 5.
3x – 8 = 5
Step 2. Collect variable terms on one side and
constants on the other side.
3x – 8 + 8 = 5 + 8
3x = 3
Solving Linear Equation
Step 3. Isolate the variable and solve.
3x = 3
3 3
x = 1
Step 4. Check the proposed solution in the original
equation by substituting 1 for x.
2(x – 4) – 5x = 5
2(1 – 4) – 5(1) = 5
10 – (5) = 5
5 = 5
Because the check results in a true statement, we
conclude that the solution set of the given equation is {1}.
Example: Solving Linear Equation
The formula relating Fahrenheit and Celsius
5
temperature scales: C = (𝐹 − 32)
9
 What is the temperature of 100° C in Fahrenheit?
5
9
100 = 𝐹 − 32
900 = 5(F – 32)
900 = 5F – 160
900 + 160 = 5F
1060
=𝐹
5
Your Turn
Solve the equations:
1. 14 – 5x = -41

14 + 41 = 5x
55 = 5x
11 = x
2. 10(3x + 2) = 70
 30x + 20 = 70
30x = 50
x = 50/30 = 5/3
3. 100 = -(x – 1) + 4(x – 6)
 100 = -x + 1 + 4x - 24
100 + 24 – 1 = 4x – x
123 = 3x
x = 41
Equations Involving Fractions
Solve:
1. 24/x = 12/7
2. (x – 2)/12 = 8/3
3. (y + 10)/10 = (y – 2)/4
(x = 14)
(x = 34)
y = 10
Application
These graphs indicate that
persons with a low sense of
humor have higher levels of
depression. These graphs can
be modeled by the following
formulas:
We are interested in the intensity of a negative life
event with an average level of depression of 7/2
for the high humor group.
1
26
D  x
9
9
7 1
26
 x
2 9
9
7
26 
1
18   18  x  
2
9 
9
7
1
26
18   18  x  18 
2
9
9
63  2 x  52
63  52  2 x
11  2 x
11 2 x

2
2
11
 x
2
Linear Equation with No Solution
Solve: 2x + 6 = 2(x + 4)
Solution:
2x + 6 = 2(x + 4)
2x + 6 = 2x + 8
2x + 6 – 2x = 2x + 8 – 2x
6=8
The original equation 2x + 6 = 2(x + 4) is equivalent
to 6 = 8, which is false for every value of x. The
equation has no solution. The solution set is Ø.
Linear Equation with Infinitely Many
solutions
Solve: 4x + 6 = 6(x + 1) – 2x
Solution:
4x + 6 = 6(x + 1) – 2x
4x + 6 = 6x + 6 – 2x
4x + 6 = 4x + 6
The original statement is equivalent to the statement
6 = 6, which is true for every value of x. The
solution set is the set of all real numbers, expressed
as
{x|x is a real number}.
Proportion
5 𝑥 3𝑥
 Ratio: , ,
8 𝑦 2𝑦
5
𝑥 3𝑥
 Proportion: =
,
8
10 2𝑦
=
4
7
Equality of two ratios
E.g.,
𝑇𝑎𝑥 𝑜𝑛 ℎ𝑜𝑢𝑠𝑒1
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 ℎ𝑜𝑢𝑠𝑒1
=
𝑇𝑎𝑥 𝑜𝑛 ℎ𝑜𝑢𝑠𝑒2
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 ℎ𝑜𝑢𝑠𝑒2
Proportion
 In many cases the amount of tax on a property is
proportional to its value.
 What does it mean?
 It means: given tax1 on value1,and tax2 on value2,
then
𝑡𝑎𝑥1
𝑡𝑎𝑥2
=
𝑣𝑎𝑙𝑢𝑒1
𝑣𝑎𝑙𝑢𝑒2
Proportions
 The property tax on a house with an assessed
value of $480,000 is $5760. Determine the
property tax on a house with an assessed value of
$600,000, assuming the same tax rate.
5760
x
------------ = -----------480,000
600,000
5760
x
------------ = -----------480,000
600,000
5760
x
----------- = -------48
60
5760 ∙ 60
------------- = x
48
5
1440 15
5760 ∙ 60
------------ = x
48
12
4
1
7200 = x
$7200 = x
Example: Changing Recipe Size
A chocolate-chip recipe for five dozen cookies requires
¾ cup of sugar. If you want to make eight dozen
cookies, how much sugar is needed?
Solution:
3 / 4 (cup)
x (cup)
----------------- = -------------5 (dozen)
8 (dozen)
8 (3/4)
---------- = x
5
8 3
6
--- ∙ --- = --- = x
5 4
5
x
60”
240”
960”
Find the height x of the tree, when a 60”-man
casts a shadow 240” long.
Proportion
x
60”
240”
60
x
----- = ------240
960
60 ∙ 960
----------- = x
240
960”
x = 240