Many equations contain variables on both sides. To solve these equations, first use the Addition or Subtraction Property of Equality to write an equivalent equation that has all of the variables on one side. Then solve.
Example
Solve each equation. Check your solution.
1. 5x = x – 12
Alternative
Solutions :
5x = x – 12 Original equation
5x – x = x – 12 – x Subtract
x from each side.
4x =
12 Simplify.
Divide
each side by 4.
x = –3 Simplify.
Check:
5x
= x – 12 Original equation
5(–3) ≟ –3 – 12 Replace x with 3.
–15 = –15 ü
You can solve an equation with a variable on both sides to
determine when a quantity that is increasing and one that is decreasing will be
the same.
Example
Sports Link
3. The 2003 World Championship 100-meter dash times were 10.07 seconds for
men and 10.85 seconds for women. If each year the men’s times decrease 0.08
second and the women’s times decrease 0.14 second, when would they have the
same winning times?
Alternative
Solutions :
Let y
represent the number of years. Write an equation.
Words 
Variables
10.07 – 0.08y 10.85 – 0.14y
Equation 10.07 – 0.08y = 10.85 – 0.14y
Write an equation.
10.07 – 0.08y + 0.14y = 10.85 – 0.14y + 0.14y
Add 0.14y to each side.
10.07 + 0.06y = 10.85
Simplify.
10.07 – 10.07 + 0.06y = 10.85 – 10.07
Subtract 10.07 from
each side.
0.06y = 0.78 Simplif
y = 13 Simplify.
At these rates, men and women have the same times 13 years after 2003, or in 2016.
Some equations have no solution. This means that there is no
value for the variable that will make the equation true. Other equations may
have every number as the solution. An equation that is true for every value of the
variable is called an identity.
Example
4. Solve each equation.
5x
+ 3 = 8 + 5x
Alternative
Solutions :
5x + 3 = 8 + 5x Original
equation
5x + 3 – 5x = 8 + 5x – 5x
Subtract 5x from each side.
3 = 8 False statement
The
equation has no solution. 3 = 8 is never
true.
5. 3 + 3t – 5 = 3t – 2
Alternative
Solutions :
3 + 3t – 5 = 3t – 2 Original equation
3t – 2 = 3t – 2 Simplify.
The
equation is an identity. 3t – 2 = 3t – 2 is true for all values of t.
Sumber
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