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2.3
Applications of Linear Equations
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Sec 2.3 - 1
2.3 Applications of Linear Equations
Problem-Solving Hint
PROBLEM-SOLVING HINT
Usually there are key words and phrases in a verbal problem that translate
into mathematical expressions involving addition, subtraction, multiplication,
and division. Translations of some commonly used expressions follow.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 2
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Addition
The sum of a number and 2
x+2
3 more than a number
x+3
7 plus a number
7+x
16 added to a number
x + 16
A number increased by 9
x+9
The sum of two numbers
x+y
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 3
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Subtraction
4 less than a number
x–4
10 minus a number
10 – x
A number decreased by 5
x–5
A number subtracted from 12
12 – x
The difference between two
numbers
x–y
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 4
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Multiplication
14 times a number
14x
A number multiplied by 8
8x
3 of a number (used with
4
fractions and percent)
3 x
4
Triple (three times) a number
3x
The product of two numbers
xy
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 5
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Division
The quotient of 6 and a number
6 (x ≠ 0)
x
A number divided by 15
x
15
The ratio of two numbers
or the quotient of two numbers
Copyright © 2010 Pearson Education, Inc. All rights reserved.
x (y ≠ 0)
y
Sec 2.3 - 6
2.3 Applications of Linear Equations
Caution
CAUTION
Because subtraction and division are not commutative operations, be careful
to correctly translate expressions involving them. For example, “5 less than a
number” is translated as x – 5, not 5 – x. “A number subtracted from 12” is
expressed as 12 – x, not x – 12.
For division, the number by which we are dividing is the denominator, and
the number into which we are dividing is the numerator. For example, “a
x . Similarly,
number divided by 15” and “15 divided into x” both translate as 15
“the quotient of x and y” is translated as xy .
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Sec 2.3 - 7
2.3 Applications of Linear Equations
Indicator Words for Equality
Equality
The symbol for equality, =, is often indicated by the word is. In fact, any
words that indicate the idea of “sameness” translate to =.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 8
2.3 Applications of Linear Equations
Translating Words into Equations
Verbal Sentence
Equation
Twice a number, decreased by 4, is 32.
If the product of a number and 16 is decreased
by 25, the result is 87.
The quotient of a number and the number plus
6 is 48.
The quotient of a number and 8, plus the
number, is 54.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 9
2.3 Applications of Linear Equations
Distinguishing between Expressions
and Equations
Decide whether each is an expression or an equation.
(a) 4(6 – x) + 2x – 1
There is no equals sign, so this is an expression.
(b) 4(6 – x) + 2x – 1 = –15
Because of the equals sign, this is an equation.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 10
2.3 Applications of Linear Equations
Six Steps to Solving Application Problems
Solving an Applied Problem
Step 1 Read the problem, several times if necessary, until you understand
what is given and what is to be found.
Step 2 Assign a variable to represent the unknown value, using diagrams
or tables as needed. Write down what the variable represents.
Express any other unknown values in terms of the variable.
Step 3 Write an equation using the variable expression(s).
Step 4 Solve the equation.
Step 5 State the answer to the problem. Does it seem reasonable?
Step 6 Check the answer in the words of the original problem.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 11
2.3 Applications of Linear Equations
Solving a Geometry Problem
The length of a rectangle is 2 ft more than three times the width. The perimeter
of the rectangle is 124 ft. Find the length and the width of the rectangle.
2.3 Applications of Linear Equations
Finding Unknown Numerical Quantities
A local grocery store baked a combined total of 912 chocolate chip cookies
and sugar cookies. If they baked 336 more chocolate chip cookies than sugar
cookies, how many of each did the store bake?
2.3 Applications of Linear Equations
Solving a Percent Problem
During a 2-day fundraiser, a local school sold 1440 raffle tickets. If they sold
350% more raffle tickets on the second day than the first day, how many raffle
tickets did they sell on the first day?
2.3 Applications of Linear Equations
Solving an Investment Problem
A local company has $50,000 to invest. It will put part of the money in an
account paying 3% interest and the remainder into stocks paying 5%. If the
total annual income from these investments will be $2180, how much will be
invested in each account?
2.3 Applications of Linear Equations
Solving a Mixture Problem
A chemist must mix 12 L of a 30% acid solution with some 80% solution to get
a 60% solution. How much of the 80% solution should be used?
2.3 Applications of Linear Equations
Problem-Solving Hint
PROBLEM-SOLVING HINT
When pure water is added to a solution, remember that water is 0% of the
chemical (acid, alcohol, etc.). Similarly, pure chemical is 100% chemical.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.3 - 17
2.3 Applications of Linear Equations
Solving a Mixture Problem
A chemist must mix 8 L of a 10% alcohol solution with pure alcohol to get a
a 40% solution. How much of the pure alcohol solution should be used?