Algebraic Expressions and Formulas

A letter in algebra stands for a number. It lets you describe an amount before you know its value. It also lets you write a rule that works with many starting numbers.

Give the letter a meaning

Suppose pens cost $2 each. Let $p$ be the number of pens. Then $2p$ gives their total cost in dollars. The letter counts pens; it does not stand for a pen itself. For 6 pens, the cost is $2\times 6=12$ dollars.

Choose a letter that helps you remember the quantity. You could use $t$ for time, but say which unit: “Let $t$ be the time in minutes.” The letter has no built-in meaning. In another problem, $t$ could count tickets. Also, $a$ does not automatically mean 1 because it comes first in the alphabet.

Unknowns, variables and general rules

A letter can have different jobs:

A variable can take different values. In the cost formula, a trip of 1 mile gives $C=7$ dollars. A trip of 4 miles gives $C=13$ dollars. Both letters change, and the formula links their values.

A constant stays fixed. The 5 in this formula is the starting fee. It stays 5 dollars while the number of miles changes. The statement $a+b=b+a$ is a generalization: adding the same two numbers in either order gives the same total. It does not ask you to find one special pair.

Keep the same meaning throughout

Every copy of a letter has the same value within one expression or problem. For $n+n+4$ with $n=3$, replace both copies:

Using 3 for the first copy and 5 for the second would change the expression. Different letters, however, can have equal values. In $x+y=10$, the pair $x=5$ and $y=5$ works. So does $x=2$ and $y=8$. Different names let us describe the two quantities separately; they do not force different numbers.

Algebra uses short ways to write familiar operations. Reading the short form correctly tells you which calculation to do.

Multiplication without a sign

$3y$ means three times $y$, and $ab$ means $a$ multiplied by $b$. A raised dot also shows multiplication: $3\cdot y=3y$. Leaving out the times sign helps keep it separate from the letter $x$.

Write the number first, followed by letters in alphabetical order. For example, $b\cdot 4\cdot a=4ab$. Multiplying in a different order leaves the value unchanged. When there is more than one number, multiply those numbers: $2\cdot a\cdot 3=6a$. A letter alone means one copy, so write $n$ for $1n$ and $-n$ for $-1n$.

Adding copies and multiplying copies

Three copies added together give $y+y+y=3y$. Repeated multiplication uses a power:

• $a\cdot a=a^2$, read “a squared.”
• $a\cdot a\cdot a=a^3$, read “a cubed.”

The raised number is the exponent. It counts how many equal factors are multiplied. The number or letter being multiplied is the base. Thus $a\cdot a\cdot b=a^{2}b$, and $x\cdot y\cdot x=x^{2}y$.

Doubling and squaring do different jobs. With $a=5$, doubling gives $2a=10$, while squaring gives $a^2=25$. In $2a$, the 2 multiplies the value. In $a^2$, the 2 tells you to multiply the value by itself.

Division and grouping

A fraction bar means divide: $\frac{n}{4}$ means “n divided by 4.” The divisor must not be zero. Half of a number can be written $\frac{n}{2}$, $\frac{1}{2}n$ or $0.5n$. Each form gives half the value of $n$.

A fraction bar also groups the whole top and bottom. Compare these with $a=6$ and $b=4$:

The first halves the whole sum. The second halves only $b$.

Parentheses also keep a group together. Read $3(n+2)$ as “add 2 to n, then multiply by 3.” With $n=4$, it gives $3(4+2)=18$. The expression $3n+2$ gives $3(4)+2=14$ instead.

Grouping changes what a power acts on, too. In $(2a{)}^2$, square the whole product. In $2a^2$, square $a$ first, then multiply by 2.

An expression describes a quantity using numbers, letters and operations. It has no equals sign. For example, $5x-2y+8$ becomes a number when you give values to its letters.

Terms, coefficients and factors

The parts added or subtracted are terms. In $5x-2y+8$, they are $5x$, $-2y$ and $8$. Keep the sign with its term.

A coefficient is the number multiplying a term's letter part. The coefficients here are 5 and −2. In $x$, the coefficient is 1. In $\frac{n}{3}=\frac{1}{3}n$, it is $\frac{1}{3}$. A term without a letter is a constant term, such as the 8 above.

A monomial is a single term made from a number and letters multiplied together, with nonnegative whole-number exponents. Examples include $5x$, $3ab$ and $2x^2$.

Factors are parts multiplied together. The term $6ab$ has factors 6, $a$ and $b$. Terms and factors name different parts: $6a+b$ has two terms, while $6ab$ is one term made from three factors.

Name the final operation

The last operation gives the name of the whole expression. In $3(x+1)$, you find a sum, then multiply it by 3. The whole expression is a product with factors 3 and $(x+1)$. In $3x+1$, addition comes last, so the whole expression is a sum.

Expressions and statements

An equation states that two expressions are equal. The equation $2x+1=9$ is true when $x=4$. Giving the letter another value can make it false.

A formula is an equation used as a rule connecting quantities. In $A=lw$, $A$ is a rectangle's area, $l$ its length and $w$ its width. The quantity alone on one side is the subject; here it is $A$.

An identity is an equation true for every allowed value of its letters. For example, $3a+2a=5a$ always holds. The sign $\equiv$ is sometimes used to mark an identity.

An inequality compares quantities with $<$, $>$, $\le$ or $\ge$. The statement $x+2>7$ says “x plus 2 is greater than 7.” It is true, for example, when $x=6$, since 8 is greater than 7.

Read a phrase as a set of actions on a starting amount. Write those actions with symbols, keeping their order and any groups.

Match words to operations

“The sum of n and 6” means $n+6$. “The product of 4 and n” means $4n$. “Twice n” is $2n$, “half of n” is $\frac{n}{2}$, and “n squared” is $n^2$. “The quotient of n and 3” means $\frac{n}{3}$.

Addition and multiplication can change order without changing the result. Thus $n+6$ and $6+n$ give the same value. Subtraction and division need more care: $n-5$ means start with $n$, while $5-n$ means start with 5.

Find the starting quantity

“5 less than n” means start with $n$ and remove 5:

“Subtract y from 5” means start with 5 and remove $y$:

The word “from” points to the starting amount. Copying the quantities in their spoken order can reverse the subtraction. To check “5 less than n,” try $n=8$. Five less than 8 is 3, and $8-5=3$. Reversing it gives −3.

“The difference of a and b” uses the named order: $a-b$. “n decreased by 7” and “7 fewer than n” both give $n-7$. For division, “n divided by 3” and “3 divided by n” also have different starting amounts.

Keep a whole sum together

“Three times the sum of a and 4” asks you to find the sum first. Put that whole sum in parentheses:

“The sum of three times a and 4” asks for multiplication first, then addition:

With $a=2$, the first gives $3(2+4)=18$. The second gives $3(2)+4=10$. The parentheses tell you which action happens first.

The same care applies when dividing. “The sum of x and y, divided by 2” is $\frac{x+y}{2}$. “x plus half of y” is $x+\frac{y}{2}$. Only the first halves both quantities together.

Read your expression back

Read $2n-3$ as “double n, then subtract 3.” Read $2(n-3)$ as “subtract 3 from n, then double.” Reading the actions aloud makes the grouping clear.

For “4 more than half of n,” write $\frac{n}{2}+4$. Try $n=10$: half of 10 is 5, and 4 more is 9. The expression gives $\frac{10}{2}+4=9$ too. A mismatch would show that the translation needs fixing.

Start by naming what changes. Choose a letter for that quantity, then build the total from the parts of the situation.

Costs and ages

Let $n$ be the number of notebooks. At $3 each, their cost is $3n$ dollars. Adding one $2 pen gives $3n+2$ dollars. The notebook price is repeated; the pen price is added once.

If four friends share the whole bill equally, each pays $\frac{3n+2}{4}$ dollars. The fraction bar groups the whole cost, including the pen. Writing $3n+\frac{2}{4}$ would split only the pen price.

Letters can also describe ages. If Sam is $s$ years old, a sister 3 years older is $s+3$. Sam's age in 4 years is $s+4$, and the age 2 years ago was $s-2$. A grandmother who is five times Sam's age is $5s$ years old.

Add the boundary to find a perimeter

A perimeter is the total distance around a shape. A rectangle has two lengths and two widths, so its perimeter is $l+w+l+w=2l+2w$. A triangle with sides $x$, $x$ and $y$ centimeters has perimeter $2x+y$ centimeters.

Side lengths can be expressions. For a triangle with sides $3x$, $x+2$ and $2x$ meters, add all three:

The result is a length in meters. Combining the copies of $x$ shortens the writing without changing the distance.

Count how a pattern is built

Picture a row of matchstick squares sharing sides. The first square uses 4 sticks. Each new square shares one side and adds 3 sticks. You can count one starting stick, then 3 for each square. For $n$ squares, that gives $3n+1$ sticks.

Check a case you can count. Three squares need $3(3)+1=10$ sticks. Counting 4 per square would count shared sides twice. The way the squares join explains why the rule keeps working.

Use more than one letter when needed

Let $a$ count apples and $b$ count bananas. The total fruit count is $a+b$. If each apple costs 40 cents and each banana costs 25 cents, the cost is $40a+25b$ cents. Each price multiplies its own count.

Check a cost expression with a small order. Tickets at $7 each with one $2 order fee give $7t+2$ dollars. One ticket costs $9; two cost $16. Those checks catch the mistake $7(t+2)$, which multiplies the fee by 7 as well.

A word formula uses quantity names to state a rule. “Pay = hours worked × hourly rate” tells you how to find pay for many different shifts.

Replace quantity names with letters

Choose a letter for each quantity and define it:

• $P$ is pay in dollars.
• $h$ is the number of hours worked.
• $r$ is the hourly rate in dollars per hour.

The word formula becomes $P=hr$. The expression $hr$ gives the pay. The whole formula links that result to its name, $P$. The subject is the quantity alone on one side, here $P$.

Separate a fixed fee from a rate

Renting a bike costs $8 plus $3 for each hour. Let $h$ be the time in hours and $C$ the cost in dollars. The repeated hourly charge is $3h$. Add the starting fee once to write $C=3h+8$.

Swapping the numbers to $8h+3$ would make 8 the hourly charge and 3 the starting fee. For 2 hours, the actual bill is $3(2)+8=14$ dollars; the swapped formula gives 19 dollars.

Write familiar measurement rules

For a rectangle, let $l$ be length and $w$ width:

• Area: $A=lw$.
• Perimeter: $P=2l+2w$.

For a rectangular box with height $h$, the volume formula is $V=lwh$. Each letter names a measurement; the subject names the quantity you calculate.

A pattern rule can become a formula too. A row of $n$ joined matchstick squares needs $3n+1$ sticks. Naming the total $S$ gives $S=3n+1$. For two squares, $S=3(2)+1=7$, which matches a direct count.

Match values to the units

The formula's units tell you how to enter a measurement. In $P=hr$, a rate per hour needs a time in hours. A 90-minute shift is $1.5$ hours, so use $h=1.5$. Entering 90 would describe 90 hours of work.

The output also has a unit. In $C=2m+5$, the cost is in dollars. In $A=lw$, lengths in centimeters give area in square centimeters.

To substitute, replace each letter with a given number. To evaluate, calculate the result. Keep every operation in place while you make the replacement.

Make hidden multiplication visible

Evaluate $5+2x$ when $x=4$:

1. Replace $x$: $5+2(4)$.
2. Multiply: $5+8$.
3. Add: $13$.

The coefficient still multiplies the value. In $3y$ with $y=4$, write $3\times 4=12$. Writing 34 would join two digits instead of multiplying.

Use the order of operations: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. PEMDAS names these groups; multiplication does not always come before division.

Work out the power first

With $x=3$, the square is $x^2=3\times 3=9$. The exponent does not mean multiply by 2.

In $2x^2$, square 3 before multiplying by 2:

In $(2x{)}^2$, the parentheses change the base:

The first squares only $x$. The second squares the whole product.

Several letters and grouped quantities

Let $a=5$, $b=2$ and $c=3$. In $a+bc$, replace each letter:

For $a^2-4b$, calculate $5^2-4(2)=25-8=17$. In $\frac{a+4}{c}$, the fraction bar groups the top, so calculate $\frac{5+4}{3}=\frac{9}{3}=3$.

Parentheses also group a difference. With $n=6$, $3(n-2)=3(6-2)=3\times 4=12$. Do the subtraction inside before multiplying.

Check a proposed equation value

Substitution can test whether an equation is true. Does $x=4$ work in $3x+2=14$? The left side gives $3(4)+2=14$. Both sides match, so 4 is a solution. With $x=5$, the left side gives 17, so 5 does not work.

You can also catch a wrong expression this way. With $n=4$, $n^2=16$ but $2n=8$. Since the results differ, squaring and doubling cannot be the same rule.

A variable can have a negative, decimal or fraction value. Replace the letter in the same way, then use the usual rules for those numbers.

Keep a negative value together

Use parentheses when substituting a negative number. With $x=-3$:

Subtracting a negative adds the matching positive number. The parentheses show that the value being removed is negative.

For a square, both factors include the sign:

The writing $-3^2$ has a different meaning: find $3^2$, then take its negative, giving −9. On a calculator, enter (−3)² to square the negative number itself.

With $a=4$ and $b=-2$, keep the same replacements throughout. Then $ab=4(-2)=-8$, while $a-b=4-(-2)=6$. For $2a^2+b$, the steps are $2(4{)}^2+(-2)=32-2=30$.

Keep the decimal point in place

With $x=1.5$, evaluate $3x+2$ as $3(1.5)+2=4.5+2=6.5$. The multiplication still comes first. With $y=0.8$, $10y=8$ and $4.2-y=3.4$.

An estimate helps catch a misplaced decimal point. The product $2.9\times 4.1$ should be near $3\times 4=12$. Its exact value, 11.89, fits that estimate.

Multiply fraction values carefully

A cube with edge length $s$ has volume $V=s^3$. For $s=\frac{1}{2}$ inch:

The volume is $\frac{1}{8}$ cubic inch. Its surface area uses $A=6s^2$. First square $\frac{1}{2}$ to get $\frac{1}{4}$, then multiply by 6: $A=\frac{6}{4}=\frac{3}{2}$ square inches. Squaring a positive fraction below 1 makes it smaller.

A fraction can also be a coefficient. With $n=12$, $\frac{3}{4}n=\frac{3}{4}\times 12=9$. Taking three quarters means divide by 4, then multiply by 3.

Check signs in division

With $a=-6$, $\frac{a}{3}+1=-2+1=-1$. With $b=-4$, $\frac{12}{b}=-3$ and $\frac{-8}{b}=2$. Matching signs give a positive quotient; different signs give a negative quotient.

Interpret the final sign using the quantity. A temperature can be negative. A negative result for the number of people or the length of a side signals a problem to check.

A formula turns measurements or counts into another quantity. Write the rule, list the known values with units, substitute, then calculate. Finish with the result's unit and meaning.

Pay from hours and rate

For pay, use $P=hr$. A 6-hour shift at $12.50 per hour gives $P=6(12.50)=75$ dollars. If the time is in minutes, change it to hours before using an hourly rate. For example, 90 minutes is 1.5 hours.

Area, perimeter and volume

For a rectangle 7 cm long and 4 cm wide:

• Area: $A=lw=7\times 4=28$ cm².
• Perimeter: $P=2l+2w=14+8=22$ cm.

Area counts square units covering a surface. Perimeter measures a boundary length. A box 5 cm long, 3 cm wide and 2 cm high has volume $V=lwh=5\times 3\times 2=30$ cm³. Volume counts cubic units filling the box.

Make all lengths use the same unit first. A rectangle 2 m long and 50 cm wide has width 0.5 m. Its area is $2\times 0.5=1$ m². Using centimeters instead gives $200\times 50=\mathrm{10,000}$ cm². Multiplying 2 by 50 before converting would mix the units.

Temperature and distance

To change Celsius temperature $C$ into Fahrenheit temperature $F$, use:

For $C=25$, first calculate $\frac{9}{5}\times 25=45$. Then add 32, giving 77 °F. For $C=-10$, the multiplication gives −18, so $F=-18+32=14$ °F. Add 32 after multiplying; the formula does not multiply the whole sum by $\frac{9}{5}$.

At a steady speed, distance equals rate multiplied by time: $d=rt$. A car traveling at 60 miles per hour for 2.5 hours covers $d=60\times 2.5=150$ miles. The time is in hours because the speed is measured per hour.

Use a rule written in words

A recipe gives “cooking time = 20 minutes per pound, plus 15 minutes.” For 6 pounds, calculate $20\times 6+15=135$ minutes. That is 2 hours 15 minutes. The extra 15 minutes is added once, after the time based on weight.

A rule takes an input, carries out operations and gives an output. Repeating the rule with different inputs helps you see how the quantities change together.

Build a table of values

For the rule “multiply by 3, then add 1,” input $n$ gives output $3n+1$. Input 5 gives $3(5)+1=16$. Changing the order to “add 1, then multiply by 3” gives $3(n+1)$ and output 18 instead.

A table of values keeps each input beside its output. For $C=2m+5$:

Each extra mile adds $2. The coefficient 2 tells you the increase for one extra input unit. The constant 5 gives the output when the input is zero.

Doubling an input

Two miles cost $9; four miles cost $13. Doubling the distance does not double the cost because the starting fee is paid only once. Compare $P=4s$: if $s$ doubles, the product doubles. There is no added fixed amount in this rule.

Change lengths one at a time

In $A=lw$, keep the width fixed to see what changing the length does. A 3 cm by 2 cm rectangle has area 6 cm². Doubling only the length gives a 6 cm by 2 cm rectangle with area 12 cm², twice as much.

Now double the width too. A 6 cm by 4 cm rectangle has area 24 cm², four times the original. Both factors doubled, so the area was multiplied by $2\times 2=4$. In $V=lwh$, doubling all three lengths multiplies the volume by $2\times 2\times 2=8$.

Let a spreadsheet repeat the rule

Put mile values in column A, starting at A2. In B2, type =2*A2+5. The star means multiply. Fill the formula down: B3 uses A3, B4 uses A4, and so on. This lets you compare many trips using one rule. Check an early row by hand to catch a typing error.

A table can also suggest a simple rule. Outputs 4, 7 and 10 for inputs 0, 1 and 2 fit $3n+4$. The outputs rise by 3 and start at 4. Those rows show a fit; a pattern's construction or a stated rule tells you whether that relationship continues.

Like terms have the same letter part: the same letters with the same exponents. Their coefficients may be different. Combining them means counting copies of that shared letter part.

Match the whole letter part

The terms $3a$ and $-7a$ are like terms. So are $5x^2$ and $x^2$, and $4ab$ and $ab$. All constant terms form a group of like terms too.

These pairs are unlike:

A matching coefficient is not enough. Look at the letters and powers before adding any coefficients.

Add copies, keeping their size

Imagine $x$ as the length of a strip. Three equal strips plus two equal strips make five strips. That is why $3x+2x=5x$. Add the coefficients and keep the letter part unchanged. Similarly, $7x-4x=3x$.

Now picture $2x+3+4x+1$ as strips of length $x$ and unit lengths. There are six strips and four units altogether, giving $6x+4$. The strips remain separate from the units because their unknown length need not be 1.

Adding copies does not create a power. The sum $x+x$ is $2x$. The product $x\cdot x$ is $x^2$. These use different operations.

Reorder without losing signs

Addition is commutative: numbers can be added in either order. It is also associative: changing how additions are grouped keeps the total. These rules let you put like terms together:

Treat a subtraction as adding a negative term. A moved term keeps its sign because subtraction itself cannot simply be reversed.

The order of factors can change too. Since $ba=ab$, the terms $2ab$ and $5ba$ are like terms. Write their common letter part as $ab$, then combine them to get $7ab$.

An expression such as $2a+3b$ stays as two terms. For $a=2$ and $b=1$, it gives 7, while $5ab$ gives 10. Adding coefficients and joining different letters would change its value.

To simplify an expression, write the same quantity more briefly. Mark every term with its sign, group like terms, then add the coefficients in each group.

Keep each term’s sign

Simplify $7a+3b-2a+b$. The $a$-terms give $7a-2a=5a$. The $b$-terms give $3b+b=4b$, since $b$ has coefficient 1. The result is $5a+4b$.

For $4x-9-6x+2$, keep the negative signs:

• Letter terms: $4x-6x=-2x$.
• Constants: $-9+2=-7$.
• Result: $-2x-7$.

In $5-2x+3x$, the coefficients are −2 and 3. They add to 1, so the result is $x+5$. Changing $-2x$ into $2x$ while regrouping would change the total.

Coefficients of zero, one and negative one

Equal positive and negative terms cancel: $3y-3y=0$. So $3y+8-3y=8$. Write no letter term when its coefficient is zero.

A coefficient of 1 is left unwritten: $4m-3m=m$. A coefficient of −1 is written with a minus sign: $2m-3m=-m$.

Powers and letter products stay in their own groups

Consider $3p^2+2pq-5+p^2-6pq+8$:

1. The $p^2$-terms give $4p^2$.
2. The $pq$-terms give $-4pq$.
3. The constants give 3.

The simplified expression is $4p^2-4pq+3$. The two letter groups cannot combine because their letter parts differ. The exponents stay unchanged when terms are added.

Decimal and fraction coefficients

The coefficient rule works for other numbers too. Subtracting gives $2.5y-0.7y=1.8y$. Adding fractions gives $\frac{1}{2}x+\frac{1}{4}x=\frac{3}{4}x$.

Remember the unwritten 1 in a lone letter. A price $p$ plus 8% sales tax is $p+0.08p=1.08p$.

Simplify a perimeter

A triangle has sides $2x+1$, $3x$ and $x+4$ centimeters. Its perimeter is $2x+1+3x+x+4=6x+5$ centimeters. With $x=3$, this gives $6(3)+5=23$ cm. The individual sides, 7, 9 and 7 cm, also total 23 cm.

The result $6x+5$ is still an expression. Simplifying has kept its value while shortening the writing; it has not found a value for $x$.

Equivalent expressions give the same value for every allowed value of their letters. They can look different while describing the same quantity.

Show equivalence by simplifying

Compare $4a+3+2a$ and $6a+1+2$. Combining like terms turns the first into $6a+3$. Combining constants turns the second into $6a+3$ too. The matching forms show that the expressions always agree.

A square's perimeter can be written $s+s+s+s$ or $4s$. Both count the same four side lengths. Another example is buying $t$ tickets at $7 on each of two days. The total $7t+7t$ dollars is equivalent to $14t$ dollars.

Joining equivalent expressions with an equals sign gives an identity. Thus $4a+3+2a=6a+3$ is true for every value of $a$. The equation $6a+3=15$ has a different job: it is true only for a value that makes both sides equal.

One mismatch is enough

To show two expressions are not equivalent, find one input giving different results. Compare $2n+3$ and $5n$ with $n=2$:

Since 7 and 10 differ, the expressions do not always agree. This input is a counterexample to the claim of equivalence.

One matching input does not prove equivalence. For $x+x$ and $x^2$, input 2 gives 4 for both. Input 3 gives 6 and 9. Their agreement at 2 was just one case. Try useful checks, then simplify to establish a rule that works for every input.

Use equivalence to save work

Suppose you need $3x+5+2x-1$ when $x=7$. Combine terms first:

Now substitute: $5(7)+4=39$. The original gives $21+5+14-1=39$ too. The shorter form needs fewer calculations.

Substitution also catches lost signs. If someone changes $5-2x+3x$ into $5-5x$, test $x=3$. The original gives $5-6+9=8$, while the proposed form gives $5-15=-10$. The mismatch identifies a wrong simplification. Keeping the signed coefficients gives $5+x$, which gives 8 as well.