4.8 Solving Equations containing Fractions
Addition Property
Let a, b, and c represent numbers. If a = b, then
+ = +
and
− =
− .
In other words, the same number may be added to or subtracted from both sides of an equation
without changing the solution of the equation.
Multiplication Property
Let a, b, and c represent numbers and let c ¹ 0. If a = b, then
∙ =
∙
and
=
In other words, both sides of an equation may be multiplied or divided by the same nonzero
number without changing the solution of the equation.
Method One: LCD Method
Step 1: If fractions are present, multiply both sides of the equation by the LCD of the
fractions.
Step 2: If parentheses are present, use the distributive property.
Step 3: Combine any like terms on each side of the equation.
Step 4: Use the addition property of equality to rewrite the equation so that variable
terms are on one side of the equation and constant terms are on the other side.
Step 5: Divide both sides of the equation by the numerical coefficient of x to solve.
Step 6: Check the answer in the original equation.
Examples: Solve.
1.
− =
2.
= +
3.
=−
4. 2 = −
Solving Equations containing Fractions applying Properties First
Solve:
To get
− =
alone, we add to both sides of the equation.
1
5
=
3 12
1 1
5 1
− + =
+
3 3 12 3
Original Equation
−
=
=
=
− =
+
∙
Multiply by
∙
Common denominator is 12
Add fractions
=
Examples: Solve.
5.
+
Add on both sides
6.
−2= −4
Simplify
7.
= −2
8.
=
+
9.3 Further Solving Linear Equations
Solving linear equations in one variable
Step 1:
If an equation contains fractions, multiply both sides by the LCD
to clear the equation of fractions.
Step 2:
Use the distributive property to remove parentheses if they are
present.
Step 3:
Simplify each side of the equation by combining like terms.
Step 4:
Get all variable terms on one side and all numbers on the other
side by using the addition property of equality.
Step 5:
Get the variable alone by using the multiplication property of
equality.
Step 6:
Check the solution by substituting it into the original equation.
Example: Solve.
1.
− =1
3. 5 − 5 = 2( + 1) + 3 − 7
2. 0.14( − 8) + 0.06 = 0.10 − 0.03(70)
4. 4(3 + 2) = 12 + 8
5. 3 − 7 = 3( + 1)
6.
−2 =
9.4: Further Problem Solving
1. UNDERSTAND the problem. During this step, become comfortable with the problem. Some
ways of doing this are:
ü Read and reread the problem.
ü Choose a variable to represent the unknown.
ü Construct a drawing.
ü Propose a solution and check it. Pay careful attention to how you check your
proposed solution. This will help when writing an equation to model the problem.
2. TRANSLATE the problem into an equation.
3. SOLVE the equation.
4. INTERPRET the results. Check the proposed solution in the stated problem and state your
conclusion.
Angles in a Triangle
The sum of the measures of the angles of any triangle is 180°.
A + B + C = 180°
Examples: Solve.
1. The measure of the second angle of a
triangle is twice the measure of the smallest
angle. The measure of the third angle of the
triangle is three times the measure of the
smallest angle. Find the measures of the
angles.
2. A 10-foot board is to be cut into two
pieces so that the length of the longer piece is
4 times the length of the shorter piece. Find
the length of each piece.
3. Some states have a single area code for the
entire state. Two such states have area codes that
are consecutive odd integers. If the sum of these
integers is 1208, find the two area codes.
9.5 Formulas and Problem Solving
A formula is an equation that states a known relationship among multiple quantities (has more
than one variable in it).
Area of a rectangle = length · width
=
=
=
=
+
=
=
9
5
simple Interest = Principal · Rate · Time
+
Perimeter of a triangle = side
ℎ
Volume of a rectangular solid = length · width · height
+ 32
distance = rate · time
Degrees in Fahrenheit =
Examples: Solve.
1. A flower bed is in the shape of a triangle
with one side twice the length of the shortest
side, and the third side is 30 feet more than
the length of the shortest side. Find the
dimensions if the perimeter is 102 feet.
+ side
+ side
9
·Degrees in Celsius + 32
5
2. The average temperature for January in
Algiers, Algeria is 59° Fahrenheit. Find the
equivalent temperature in Celsius.
Solving a Formula for a Variable
It is often necessary to rewrite a formula so that it is solved for one of the variables.
To solve a formula or an equation for a specified variable, we use the same steps as for solving a
linear equation except that we treat the specified variable as the only variable in the equation.
Solving a Formula for a Specified Variable
Step 1: Multiply on both sides to clear the equation of fractions if they
appear.
Step 2: Use the distributive property to remove parentheses if they
appear.
Step 3: Simplify each side of the equation by combining like terms.
Step 4: Get all terms containing the specified variable on one side and
all other terms on the other side by using the addition property
of equality.
Step 5: Get the specified variable alone by using the multiplication
property of equality.
Examples: Solve for the indicated variable.
3.
= 2 ; solve for
5.
=
; solve for
4.
=
+
+ ; solve for
6. 4 + 3 = 9; solve for