In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into
algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in Table \(\PageIndex{3}\). a plus b the sum of a and b a increased by b b more than a the total of a and b b added to a a minus b the difference of a and b b subtracted from a a decreased by b b less than a a times b the product of a and b a divided by b the quotient of a and b the ratio of a and b b divided into a Look closely at these phrases using the four operations: Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers. Translate each word phrase into an algebraic expression: Solution the difference of \(20\) and \(4\) \(20\) minus \(4\) \(20 − 4\)
the quotient of \(10x\) and \(3\) divide \(10x\) by \(3\) \(10x ÷ 3\) This can also be written as 1\(0x / 3\) or \(\dfrac{10x}{3}\) exercise \(\PageIndex{21}\)Translate the given word phrase into an algebraic expression:
\(47-41\) Answer b\(5x\div 2\) exercise \(\PageIndex{22}\)Translate the given word phrase into an algebraic expression:
\(17+19\) Answer b\(7x\) How old will you be in eight years? What age is eight more years than your age now? Did you add \(8\) to your present age? Eight more than means eight added to your present age. How old were you seven years ago? This is seven years less than your age now. You subtract \(7\) from your present age. Seven less than means seven subtracted from your present age. Example \(\PageIndex{12}\): translateTranslate each word phrase into an algebraic expression:
Solution
Eight more than \(y\) Eight added to \(y\) \(y + 8\)
Seven less than \(9z\) Seven subtracted from \(9z\) \(9z − 7\) exercise \(\PageIndex{23}\)Translate each word phrase into an algebraic expression:
\(x+11\) Answer b\(11a-14\) exercise \(\PageIndex{24}\)Translate each word phrase into an algebraic expression:
\(j+19\) Answer b\(2x-21\) Example \(\PageIndex{13}\): translateTranslate each word phrase into an algebraic expression:
Solution
five times the sum of \(m\) and \(n\) \(5(m + n)\)
the sum of five times \(m\) and \(n\) \(5m + n\) Notice how the use of parentheses changes the result. In part (a), we add first and in part (b), we multiply first. exercise \(\PageIndex{25}\)Translate the word phrase into an algebraic expression:
\(4(p+q)\) Answer b\(4p+q\) exercise \(\PageIndex{26}\)Translate the word phrase into an algebraic expression:
\(2x-8\) Answer b\(2(x-8)\) Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples. Example \(\PageIndex{14}\): write an expressionThe height of a rectangular window is 6 inches less than the width. Let w represent the width of the window. Write an expression for the height of the window. Solution
exercise \(\PageIndex{27}\)The length of a rectangle is \(5\) inches less than the width. Let \(w\) represent the width of the rectangle. Write an expression for the length of the rectangle. Answer\(w-5\) exercise \(\PageIndex{28}\)The width of a rectangle is \(2\) meters greater than the length. Let \(l\) represent the length of the rectangle. Write an expression for the width of the rectangle. Answer\(l+2\) Example \(\PageIndex{15}\): write an expressionBlanca has dimes and quarters in her purse. The number of dimes is \(2\) less than \(5\) times the number of quarters. Let \(q\) represent the number of quarters. Write an expression for the number of dimes. Solution
exercise \(\PageIndex{29}\)Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let \(q\) represent the number of quarters. Write an expression for the number of dimes. Answer\(6q-7\) exercise \(\PageIndex{30}\)Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let \(n\) represent the number of nickels. Write an expression for the number of dimes. Answer\(4n+8\) Key Concepts
GlossarytermA term is a constant or the product of a constant and one or more variables. coefficientThe constant that multiplies the variable(s) in a term is called the coefficient. Terms that are either constants or have the same variables with the same exponents are like terms. evaluateTo evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. Practice Makes PerfectEvaluate Algebraic ExpressionsIn the following exercises, evaluate the expression for the given value.
Identify Terms, Coefficients, and Like TermsIn the following exercises, list the terms in the given expression.
In the following exercises, identify the coefficient of the given term.
In the following exercises, identify all sets of like terms.
Simplify Expressions by Combining Like TermsIn the following exercises, simplify the given expression by combining like terms.
Translate English Phrases into Algebraic ExpressionsIn the following exercises, translate the given word phrase into an algebraic expression.
In the following exercises, write an algebraic expression.
Everyday MathIn the following exercises, use algebraic expressions to solve the problem.
Writing Exercises
Self Check(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. (b) After reviewing this checklist, what will you do to become confident for all objectives? Contributors and Attributions
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