Page 620 - Basic College Mathematics with Early Integers

S E C T I O N 8.5 I EQUATIONS AND PROBLEM SOLVING 597

Objective Using Problem-Solving Steps to Solve Problems

Our main purpose for studying arithmetic and algebra is to solve problems. The
same problem-solving steps that have been used throughout this text are used in
this section also.Those steps are next.

Problem-Solving Steps

1. UNDERSTAND the problem. During this step, become comfortable with
the problem. Some ways of doing this are as follows:
• Read and reread the problem.
• Construct a drawing.
• Propose a solution and check. Pay careful attention to how you check
your proposed solution.This will help when writing an equation to model
the problem.
• Choose a variable to represent an unknown. Use this variable to repre-
sent any other unknowns.
2. TRANSLATE the problem into an equation.
3. SOLVE the equation.
4. INTERPRET the results: Check the proposed solution in the stated prob-
lem and state your conclusion.

The first problem that we solve consists of finding an unknown number.

Example 3 Finding an Unknown Number PRACTICE 3
Translate “The difference of a
Twice a number plus 3 is the same as the number minus 6.Find the unknown number.
number and 2 equals 6 added
to three times the number” into
Solution:
an equation and solve.
1. UNDERSTAND the problem.To do so, we read and reread the problem.
Let’s propose a solution to help us understand. Suppose the unknown num-
#
ber is 5. Twice this number plus 3 is 2 5 + 3 or 13. Is this the same as the num-
ber minus 6, or 5 - 6, or -1? Since 13 is not the same as -1, we know that 5 is
not the solution. However, remember that the purpose of proposing a solution
is not to guess correctly, but to better understand the problem.
Now let’s choose a variable to represent the unknown. Let’s let
x = unknown number
2. TRANSLATE the problem into an equation.
In words:
twice a is the the number
plus 3
number same as minus 6
T T T T
Translate: 2x + 3 = x - 6
3. SOLVE the equation.To solve the equation, we first subtract x from both sides.
2x + 3 = x - 6
2x + 3 - x = x - 6 - x
x + 3 =-6 Simplify.
x + 3 - 3 =-6 - 3 Subtract 3 from both sides.
Answer
x =-9 Simplify. 3. x - 2 = 6 + 3x; -4
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