Expressions

Study Sheet

Expressions

Letters standing for numbers

Substituting a Value

Concept
The Big Idea

A letter holds a number we have not chosen yet. To evaluate, replace the letter with its value and follow the order of operations.

A variable is a letter standing for a number we do not know yet, or that can change. An expression like 3x+23x + 2 is a recipe: triple the number, then add two. Evaluating means substituting a value; simplifying means combining like terms — terms with the same variable part — because 3x+2x3x + 2x is five of the same thing.

Example
Evaluating 3x+53x + 5 at x=4x = 4
x12343x + 58111417
The same recipe for each x; at x = 4 it gives 17.

Substitute: 3(4)+5=12+5=173(4) + 5 = 12 + 5 = 17.

Example
Evaluating 2x22x^2 at x=3x = 3

Powers come before multiplication: x2=32=9x^2 = 3^2 = 9, then 2×9=182 \times 9 = 18. Doubling first, (2×3)2=36(2 \times 3)^2 = 36, would be a different expression.

Side note
Where it goes wrong

Combining unlike terms: 3x+23x + 2 is NOT 5x5x. Only terms with identical variable parts can be added, the way 33 apples and 22 oranges stay separate.

Try it
Try it: substitute

Evaluate 6y−26y - 2 at y=3y = 3, and k4+1\dfrac{k}{4} + 1 at k=20k = 20.

Show the answer

6(3)−2=18−2=166(3) - 2 = 18 - 2 = \mathbf{16}. 204+1=5+1=6\dfrac{20}{4} + 1 = 5 + 1 = \mathbf{6}.

In Plain Terms & More Examples

Concept
In plain terms

A letter stands for a number. To evaluate, swap in the value and follow the order of operations.

Example
Evaluating 2x+72x + 7 at x=5x = 5

Substitute: 2(5)+7=10+7=172(5) + 7 = 10 + 7 = 17.

Example
Evaluating 5a−35a - 3 at a=4a = 4

Substitute: 5(4)−3=20−3=175(4) - 3 = 20 - 3 = 17.

Side note
Where it goes wrong

Writing the value next to the number as digits. 5a5a at a=4a = 4 means 5×4=205 \times 4 = 20, not 5454. Put the value in brackets when you substitute: 5(4)5(4).

Try it
Try it: two more

Evaluate 12−2p12 - 2p at p=3p = 3, and p2+1p^2 + 1 at p=3p = 3.

Show the answer

12−2(3)=12−6=612 - 2(3) = 12 - 6 = \mathbf{6}. 32+1=9+1=103^2 + 1 = 9 + 1 = \mathbf{10}.

Distributing and Factoring

Concept
Expand or pull out
3xx1243
One rectangle, two pieces: 3x + 12.

3(x+4)=3x+123(x + 4) = 3x + 12 by distributing. Backwards, 6x+9=3(2x+3)6x + 9 = 3(2x + 3) by pulling out the common factor 33.

Why does 3(x+4)3(x + 4) equal 3x+123x + 12? Draw a rectangle 33 tall and x+4x + 4 wide. Cut it where the xx ends: the two pieces are 33 by xx (area 3x3x) and 33 by 44 (area 1212). Together they are the whole rectangle, so 3(x+4)=3x+123(x + 4) = 3x + 12 for every value of xx.

Example
Simplify 2(x+5)+3x−42(x + 5) + 3x - 4

Distribute: 2x+10+3x−42x + 10 + 3x - 4. Combine: 5x+65x + 6.

Example
Factoring 8x+128x + 12
8x2x1234
Factoring finds the side lengths from the pieces.

Both terms are multiples of 44: 8x=4×2x8x = 4 \times 2x and 12=4×312 = 4 \times 3. So 8x+12=4(2x+3)8x + 12 = 4(2x + 3). Check by distributing: 4×2x+4×3=8x+124 \times 2x + 4 \times 3 = 8x + 12.

Side note
Where it goes wrong

Distributing to the first term only. 3(x+4)3(x + 4) is not 3x+43x + 4: the 33 multiplies everything in the bracket, both the xx and the 44. The rectangle has two pieces, 3x3x and 1212.

Try it
Try it: simplify and evaluate

Simplify 4(2y−1)+y4(2y - 1) + y, then evaluate it at y=3y = 3.

Show the answer

8y−4+y=9y−48y - 4 + y = \mathbf{9y - 4}. At y=3y = 3: 27−4=2327 - 4 = \mathbf{23}.

Writing Expressions

Concept
Phrases into expressions

More than and increased by add; less than and decreased by subtract; times and of multiply. Careful: "55 less than nn" is n−5n - 5, so the order flips.

"55 less than nn" means start at nn and go down 55, so nn comes first. Testing with a number catches order mistakes: 55 less than 1212 is 77, and 12−5=712 - 5 = 7, while 5−125 - 12 is negative.

Example
Perimeter
2w + 3w2w + 3w
Two widths and two lengths make the perimeter.

A rectangle has width ww and length 2w+32w + 3. Its perimeter is 2w+2(2w+3)=6w+62w + 2(2w + 3) = 6w + 6.

Example
"5 less than a number"

Let the number be nn. Five less than nn is n−5n - 5. Test it: if n=12n = 12, five less is 77, and 12−5=712 - 5 = 7.

Side note
Where it goes wrong

Dropping the brackets. "Twice the sum of xx and 33" is 2(x+3)=2x+62(x + 3) = 2x + 6. Writing 2x+32x + 3 doubles only the xx. Test with x=1x = 1: twice the sum of 11 and 33 is 88, but 2(1)+3=52(1) + 3 = 5.

Try it
Try it: the pattern
Three tables in a row, with a dot for each seat.

Each table seats 44 people, but joining nn tables in a row loses 22 seats per join. Write the seats for nn tables in a row, and evaluate for n=5n = 5.

Show the answer

There are n−1n - 1 joins: 4n−2(n−1)=2n+24n - 2(n - 1) = \mathbf{2n + 2} seats. For n=5n = 5: 12\mathbf{12}.

Combining Like Terms

Concept
Count the same things together

Like terms have exactly the same letter part, like 4a4a and 2a2a. Add or subtract their numbers and keep the letter: 4a+2a=6a4a + 2a = 6a. Unlike terms, like 6a6a and 2b2b, stay separate.

4a+2a4a + 2a is four of something plus two more of the same thing: six of it. That is the distributive property read backwards, (4+2)a(4 + 2)a. Nothing joins aa's with bb's, just as 33 apples and 22 oranges do not make 55 of either.

Example
Simplifying 4a+3b+2a−b4a + 3b + 2a - b

Group the like terms: (4a+2a)+(3b−b)=6a+2b(4a + 2a) + (3b - b) = 6a + 2b. The −b-b belongs with the bb's: 3b−1b=2b3b - 1b = 2b.

Example
Simplifying 5x+3−2x+75x + 3 - 2x + 7

xx-terms: 5x−2x=3x5x - 2x = 3x. Numbers: 3+7=103 + 7 = 10. Result: 3x+103x + 10.

Example
Checking with a number

Put x=2x = 2 into both forms. Original: 5(2)+3−2(2)+7=10+3−4+7=165(2) + 3 - 2(2) + 7 = 10 + 3 - 4 + 7 = 16. Simplified: 3(2)+10=163(2) + 10 = 16. They agree, as equal expressions must for every xx.

Side note
Where it goes wrong

Multiplying the letters when adding. 3x+2x3x + 2x is 5x5x, not 5x25x^2. Adding changes how many xx's there are, not what is being counted: three xx's and two xx's are five xx's.

Try it
Try it: simplify and check

Simplify 7m+4−3m+17m + 4 - 3m + 1, then check your answer at m=2m = 2.

Show the answer

7m−3m=4m7m - 3m = 4m and 4+1=54 + 1 = 5: 4m+5\mathbf{4m + 5}. Check at m=2m = 2: the original gives 14+4−6+1=1314 + 4 - 6 + 1 = 13, and 4(2)+5=134(2) + 5 = \mathbf{13}.

Why It Works

Like terms combine because of the distributive property run backwards: 3x+2x=(3+2)x=5x3x + 2x = (3 + 2)x = 5x. Unlike terms have no common factor to pull out, so they stay apart. Every simplification rule you will learn — collecting terms, expanding brackets, factoring — is the distributive property a(b+c)=ab+aca(b + c) = ab + ac read in one direction or the other.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.