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Two expressions that agree for every value, and how to show it.
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In this lesson you decide whether two expressions are equivalent, meaning they give the same value for every input, not just for the one you tried. Combining like terms, using the fact that a sum can be reordered and regrouped, shows it. A single matching value does not, and understanding why one example is not enough is the beginning of understanding proof.
You can write an expression such as $4y - 9$ and evaluate it when you are told the value of the letter. You know that $3y$ means 3 times $y$, that a lone $y$ means $1y$, and that numbers can be added in any order: $8 + 5$ and $5 + 8$ are both 13. In this lesson you learn when two different-looking expressions are really the same, and how to rewrite an expression in a shorter form without changing what it means.
| Term | What it means |
|---|---|
| Equivalent expressions | Expressions that give the same value for every value of the variable, such as $y + y + y$ and $3y$. |
| Like terms | Terms with exactly the same variable part, such as $5x$ and $2x$, or two plain numbers. |
| Unlike terms | Terms with different variable parts, such as $5x$ and 5, or $5x$ and $5y$. They cannot be combined. |
| Combine like terms | Add or subtract the coefficients of like terms to write them as one term: $5x + 2x = 7x$. |
| Simplify | Rewrite an expression as an equivalent one with as few terms as possible. |
| Commutative property | Changing the order of an addition or a multiplication does not change the answer: $a + b = b + a$. |
| Associative property | Changing the grouping of an addition or a multiplication does not change the answer: $(a + b) + c = a + (b + c)$. |
| Counterexample | One value that makes two expressions disagree, which shows they are not equivalent. |
Two expressions are equivalent when they give the same value no matter what number you put in for the variable. $y + y + y$ and $3y$ are equivalent: three $y$'s added together is 3 times $y$, whatever $y$ is. Try $y = 4$: $4 + 4 + 4 = 12$ and $3 \times 4 = 12$. Try $y = 100$: both give 300. They are two ways of writing the same thing.
Equivalent does not mean 'equal for the number I tried'. The expressions $2x + 3$ and $5x$ both give 5 when $x = 1$. But at $x = 2$ the first gives 7 and the second gives 10. So they are not equivalent. They just happened to meet at one value. A single agreement proves nothing; a single disagreement proves they are different.
How, then, can we be sure two expressions agree for every value, when there are infinitely many values to try? We use the properties of operations, rules that are true for all numbers. If each step of a rewrite follows one of those rules, the result is equivalent for every value at once. The main tool in this lesson is combining like terms: $5x + 2x = 7x$, because 5 of anything plus 2 more of the same thing is 7 of it.
Another way: picture
Picture $x$ as a bag with an unknown number of marbles inside. $5x + 2x$ is 5 bags and then 2 more bags: 7 bags, however many marbles each holds. But $5x + 2$ is 5 bags and 2 loose marbles. You cannot call that 7 bags, because 2 loose marbles are not 2 bags.
Another way: story
Your family buys 3 bags of apples on Monday and 2 bags on Thursday, and every bag holds the same number of apples, $a$. Counting bag by bag gives $3a + 2a$; counting all the bags at once gives $5a$. Both count the same apples, so the expressions are equivalent.
Like terms have exactly the same variable part. $6m$ and $m$ are like terms (the lone $m$ is $1m$). The numbers 4 and 11 are like terms with each other. But $6m$ and 6 are unlike, and so are $6m$ and $6n$, because $m$ and $n$ may stand for different numbers.
Only like terms combine. To combine them, add or subtract their coefficients and keep the variable part the same:
$$6m + m = 7m \qquad 9n - 4n = 5n \qquad 4 + 11 = 15$$
The letter does not change when you combine. $6m + m$ is $7m$, not $7m^2$. You are counting how many $m$'s there are, just as 6 apples plus 1 apple is 7 apples, not 7 'square apples'.
An expression is simplified when no two of its terms are alike. $7m + 15$ is simplified: one $m$ term and one number. Stopping earlier, such as $6m + m + 15$, is still correct and still equivalent, just longer.
Before you can combine like terms, you often need to move them next to each other. Two properties make that safe.
Multiplication has the same two properties: $4 \times x = x \times 4$, and $2 \times (3x) = (2 \times 3)x = 6x$.
Subtraction does not have them: $9 - 4$ is 5 but $4 - 9$ is not. So when you move a term, carry the sign in front of it along, as part of the term. In $8x + 5 - 3x$ the third term is '$-3x$'. Moving it gives $8x - 3x + 5$, and then $5x + 5$. Thinking of subtraction as adding a negative keeps you safe: $8x + 5 + (-3x)$ can be rearranged like any sum.
Substituting a number into both expressions is a fast way to check your work, and a reliable way to catch a mistake.
Say you simplified $4a + 7 + 2a$ to $6a + 7$. Pick $a = 3$. The original gives $12 + 7 + 6 = 25$; your answer gives $18 + 7 = 25$. They agree, which makes you more confident. If you had written $13a$ by mistake, $a = 3$ would give 39, and the disagreement would show the error at once.
Choose test values with care. Avoid 0 and 1: many wrong answers agree with right ones there, because multiplying by 1 changes nothing and multiplying by 0 wipes everything out. A value like 3, 5 or 10 is better. Two different values are better still.
Remember what each result means. A disagreement proves the expressions are not equivalent; that value is a counterexample. Agreement only suggests they are. The proof that they agree for every value comes from the properties you used to rewrite one into the other.
When an expression has two different letters, sort the terms by letter. $3p + 2q + 5p + q$ becomes $3p + 5p + 2q + q$, and then $8p + 3q$. That is as simple as it gets. You cannot add $8p$ and $3q$ to get $11pq$ or $11p$, because $p$ and $q$ may be different numbers. If $p$ is the price of a pizza and $q$ the price of a quart of juice, 8 pizzas and 3 quarts of juice are not 11 of anything.
The same goes for a term with a letter and a plain number. $8p + 3$ stays as it is: 8 pizzas and 3 extra dollars.
Step 1: Find the terms. Split the expression at its plus and minus signs, keeping each sign with the term after it. Rewrite any lone letter with coefficient 1.
Step 2: Sort the like terms. Move terms with the same variable part next to each other, and the plain numbers together. The commutative and associative properties allow this.
Step 3: Combine. Add or subtract the coefficients of each group of like terms. Keep the variable part the same.
Step 4: Write the result with one term for each kind, usually letter terms first and the constant last.
Step 5: Check by substituting a value such as 3 into the original and into your answer. The values must match. If they do not, go back and look for a dropped sign, a lone letter counted as 0 instead of 1, or a number combined with a letter term.
To decide whether two given expressions are equivalent, simplify both. If they simplify to the same thing, they are equivalent. If they do not, find a counterexample to be sure.
An NBA basketball court is a rectangle 94 feet long and 50 feet wide. How much rope would it take to go all the way around it? The perimeter of a rectangle with length $l$ and width $w$ can be written in more than one way:
$$l + w + l + w \qquad\text{or}\qquad 2l + 2w$$
The two are equivalent: reorder the first to $l + l + w + w$, then combine like terms. Evaluate either one with $l = 94$ and $w = 50$: $94 + 50 + 94 + 50 = 288$, and $2(94) + 2(50) = 188 + 100 = 288$ feet.
A high school court is shorter, 84 feet by 50 feet, and the same equivalent expression gives $168 + 100 = 268$ feet. Builders, painters and sports-field crews choose whichever form is quicker to work out. Knowing they are equivalent means the choice can never change the answer.
A snack bar sells hot dogs for $h$ dollars each and bottles of water for $b$ dollars each. During a Little League game it sells 38 hot dogs and 45 waters in the first half, and 27 hot dogs and 52 waters in the second half. The total sales are
$$38h + 45b + 27h + 52b = 65h + 97b$$
dollars, after combining like terms. The simplified form is much easier to use. If hot dogs cost 3 dollars and water 2 dollars, the snack bar took in $65(3) + 97(2) = 195 + 194 = 389$ dollars. Store computers and spreadsheets do the same thing: they total each item separately before multiplying by its price, because hot dogs and water are unlike terms with different prices.
Combining unlike terms. $4x + 3$ is not $7x$. The 3 is a plain number, not 3 $x$'s.
Forgetting a lone letter. In $5y + y$, the second $y$ is $1y$, so the sum is $6y$, not $5y$.
Changing the letter. $2n + 3n = 5n$, not $5n^2$. Adding $n$'s gives more $n$'s; only multiplying $n$ by $n$ gives $n^2$.
Losing a minus sign. In $9k + 2 - 4k$, the $4k$ is subtracted, so the answer is $5k + 2$, not $13k + 2$. Move each term with the sign in front of it.
Trusting one matching value. Two expressions that agree at $x = 1$ may still be different. Test with another value, and avoid 0 and 1.
Mixing different letters. $3a + 4b$ does not simplify. $a$ and $b$ may be different numbers.
Is $y + y + y$ equivalent to $3y$? Say what each one means.
$y + y + y = \text{three } y\text{'s added}, \quad 3y = 3 \times y$
Reading an expression in words is the first step to comparing it with another.
Use the meaning of multiplication.
$3 \times y = y + y + y$
Multiplying by 3 means adding three copies, for any number $y$.
Check with $y = 4$.
$4 + 4 + 4 = 12, \qquad 3 \times 4 = 12$
A check with a number catches a mistake if there is one.
Check with $y = 10$.
$10 + 10 + 10 = 30, \qquad 3 \times 10 = 30$
A second value, well away from the first, makes the check stronger.
State the conclusion.
$y + y + y \equiv 3y$
They are equivalent because of what multiplication means; the checks only agree with that.
Simplify $7a + 4 + 2a - 1 + a$. Write the lone $a$ as $1a$.
$7a + 4 + 2a - 1 + 1a$
A letter on its own has coefficient 1.
Move the $a$ terms together, keeping each sign with its term.
$7a + 2a + 1a + 4 - 1$
Terms of a sum can be reordered; the minus belongs to the 1.
Add the coefficients of $a$.
$7 + 2 + 1 = 10, \text{ so } 10a$
7, 2 and 1 $a$'s make 10 $a$'s.
Combine the constants.
$4 - 1 = 3$
Plain numbers are like terms with each other.
Write the simplified expression.
$10a + 3$
No two terms are alike now, so this is as simple as it gets.
Check both forms with $a = 2$.
$14 + 4 + 4 - 1 + 2 = 23, \qquad 20 + 3 = 23$
The values agree, so no sign or term was lost.
Is $2x + 3$ equivalent to $5x$? Try $x = 1$ in the first expression.
$2(1) + 3 = 5$
Testing with a value is a quick way to compare.
Try $x = 1$ in the second expression.
$5(1) = 5$
Both give 5, so this value does not tell them apart.
Try another value, $x = 2$, in the first expression.
$2(2) + 3 = 7$
One agreement is not enough, so test again.
Try $x = 2$ in the second expression.
$5(2) = 10$
Substitute the same value into both.
Compare the results.
$7 \ne 10$
The expressions disagree at $x = 2$.
State the conclusion.
$2x + 3 \not\equiv 5x$
One disagreement is a counterexample, and it proves they are not equivalent.
Explain the mistake behind the claim.
$2x \text{ and } 3 \text{ are unlike terms}$
The 3 is a plain number, not 3 $x$'s, so it cannot join the $2x$.
Put the $b$ terms together.
$6b + 3b + 5$
The terms of a sum can be reordered.
Add the coefficients of $b$.
$6b + 3b = 9b$
6 $b$'s and 3 more $b$'s make 9 $b$'s.
Write the simplified expression.
Which expression is equivalent to $8y + 3 + 7y$?
A triangular flower bed has sides of $5x + 9$ feet, $2x$ feet and $x + 12$ feet. Complete the solution to write its perimeter as simply as possible, then find the perimeter when $x = 11$.
Add the three sides.
$P = (5x + 9) + 2x + (x + 12)$
The perimeter is the distance all the way around.
Put the like terms side by side.
$P = 5x + 2x + x + 9 + 12$
In a sum the terms may be reordered and regrouped without changing the value.
Add the coefficients of $x$.
$5 + 2 + 1 =$ coef
The lone $x$ adds 1 to the count of $x$'s.
Add the constants.
$9 + 12 =$ const
The plain numbers are like terms with each other.
Evaluate the simplified perimeter at $x = 11$.
$P =$ p feet
The simplified expression is equivalent to the sum of the sides, so it gives the same perimeter with less work.
Simplify $9x + 4 + x - 5x + 14$.
Answer:
Riley says $2x + 9$ is equivalent to $11x$. Test the claim with $x = 8$: find the value of each expression.
$2x + 9$ gives first and $11x$ gives second.
Fill in the blanks so that the two sides are equivalent: $8k + 16 + \square k + \square = 14k + 18$.
The missing coefficient is coef and the missing constant is const.
A school store orders supplies twice. In August it buys 3 boxes of notebooks and 5 boxes of pens. In January it buys 7 boxes of notebooks and 4 boxes of pens. A box of notebooks costs $n$ dollars and a box of pens costs $p$ dollars. Write the total cost of both orders as simply as you can.
Answer:
A rectangular dog run is $x + 16$ feet long and $x$ feet wide. Write an expression for the length of fence needed to go all the way around it, as simply as you can.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A school store orders supplies twice. In August it buys 6 boxes of notebooks and 5 boxes of pens. In January it buys 8 boxes of notebooks and 8 boxes of pens. A box of notebooks costs $n$ dollars and a box of pens costs $p$ dollars. Write the total cost of both orders as simply as you can.
Answer:
You can decide whether two expressions are equivalent. Without looking: why is checking $x = 2$ not enough to show two expressions are equivalent?
16. Your turn: simplify $6b + 5 + 3b$, step 3
$9b + 5$
The 5 has no $b$, so it stays as its own term.