# 1.2 Use the language of algebra  (Page 5/18)

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Evaluate $3{x}^{2}+4x+1$ when $x=3.$

40

Evaluate $6{x}^{2}-4x-7$ when $x=2.$

9

## Indentify and combine like terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.

## Term

A term    is a constant, or the product of a constant and one or more variables.

Examples of terms are $7,y,5{x}^{2},9a,\text{and}\phantom{\rule{0.2em}{0ex}}{b}^{5}.$

The constant that multiplies the variable is called the coefficient .

## Coefficient

The coefficient    of a term is the constant that multiplies the variable in a term.

Think of the coefficient as the number in front of the variable. The coefficient of the term 3 x is 3. When we write x , the coefficient is 1, since $x=1·x.$

Identify the coefficient of each term: 14 y $15{x}^{2}$ a .

## Solution

The coefficient of 14 y is 14.

The coefficient of $15{x}^{2}$ is 15.

The coefficient of a is 1 since $a=1\phantom{\rule{0.2em}{0ex}}a.$

Identify the coefficient of each term: $17x$ $41{b}^{2}$ z .

14 41 1

Identify the coefficient of each term: 9 p $13{a}^{3}$ ${y}^{3}.$

9 13 1

Some terms share common traits. Look at the following 6 terms. Which ones seem to have traits in common?

$\begin{array}{cccccccccccccccc}5x\hfill & & & 7\hfill & & & {n}^{2}\hfill & & & 4\hfill & & & 3x\hfill & & & 9{n}^{2}\hfill \end{array}$

The 7 and the 4 are both constant terms.

The 5x and the 3 x are both terms with x .

The ${n}^{2}$ and the $9{n}^{2}$ are both terms with ${n}^{2}.$

When two terms are constants or have the same variable and exponent, we say they are like terms .

• 7 and 4 are like terms.
• 5 x and 3 x are like terms.
• ${x}^{2}$ and $9{x}^{2}$ are like terms.

## Like terms

Terms that are either constants or have the same variables raised to the same powers are called like terms    .

Identify the like terms: ${y}^{3},$ $7{x}^{2},$ 14, 23, $4{y}^{3},$ 9 x , $5{x}^{2}.$

## Solution

${y}^{3}$ and $4{y}^{3}$ are like terms because both have ${y}^{3};$ the variable and the exponent match.

$7{x}^{2}$ and $5{x}^{2}$ are like terms because both have ${x}^{2};$ the variable and the exponent match.

14 and 23 are like terms because both are constants.

There is no other term like 9 x .

Identify the like terms: $9,$ $2{x}^{3},$ ${y}^{2},$ $8{x}^{3},$ $15,$ $9y,$ $11{y}^{2}.$

9 and 15, ${y}^{2}$ and $11{y}^{2},$ $2{x}^{3}$ and $8{x}^{3}$

Identify the like terms: $4{x}^{3},$ $8{x}^{2},$ 19, $3{x}^{2},$ 24, $6{x}^{3}.$

19 and 24, $8{x}^{2}$ and $3{x}^{2},$ $4{x}^{3}$ and $6{x}^{3}$

Adding or subtracting terms forms an expression. In the expression $2{x}^{2}+3x+8,$ from [link] , the three terms are $2{x}^{2},3x,$ and 8.

Identify the terms in each expression.

1. $9{x}^{2}+7x+12$
2. $8x+3y$

## Solution

The terms of $9{x}^{2}+7x+12$ are $9{x}^{2},$ 7 x , and 12.

The terms of $8x+3y$ are 8 x and 3 y .

Identify the terms in the expression $4{x}^{2}+5x+17.$

$4{x}^{2},5x,17$

Identify the terms in the expression $5x+2y.$

5 x , 2 y

If there are like terms in an expression, you can simplify the expression by combining the like terms. What do you think $4x+7x+x$ would simplify to? If you thought 12 x , you would be right!

$\begin{array}{c}\hfill 4x+7x+x\hfill \\ \hfill x+x+x+x\phantom{\rule{1em}{0ex}}+x+x+x+x+x+x+x\phantom{\rule{1em}{0ex}}+x\hfill \\ \hfill 12x\hfill \end{array}$

Add the coefficients and keep the same variable. It doesn’t matter what x is—if you have 4 of something and add 7 more of the same thing and then add 1 more, the result is 12 of them. For example, 4 oranges plus 7 oranges plus 1 orange is 12 oranges. We will discuss the mathematical properties behind this later.

Simplify: $4x+7x+x.$

## How to combine like terms

Simplify: $2{x}^{2}+3x+7+{x}^{2}+4x+5.$

## Solution

Simplify: $3{x}^{2}+7x+9+7{x}^{2}+9x+8.$

$10{x}^{2}+16x+17$

Simplify: $4{y}^{2}+5y+2+8{y}^{2}+4y+5.$

$12{y}^{2}+9y+7$

## Combine like terms.

1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add or subtract the coefficients and keep the same variable for each group of like terms.

Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
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Mckenzie
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Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes