# 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.

4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
divide 3x⁴-4x³-3x-1 by x-3
how to multiply the monomial
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9
Did anyone else have trouble getting in quiz link for linear inequalities?
operation of trinomial