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Linear expressions, two-step equations and inequalities

Rewriting linear expressions: expanding, subtracting brackets, factoring and percent multipliers.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

In this lesson you rewrite linear expressions into equivalent forms. You expand brackets by multiplying every term inside, subtract a bracket by flipping every sign in it, collect like terms, and factor out the greatest common factor. You also write a percent increase or decrease as a single multiplier, and you check every rewrite by putting in a number for the letter. Simplifying an expression is the first step in solving most equations and inequalities.

2. What you already know

You know the distributive property for numbers: $3 \times 24$ is $3 \times 20 + 3 \times 4 = 60 + 12 = 72$. You can use a letter to stand for a number, so $5n$ means 5 times $n$. You can add, subtract and multiply positive and negative numbers, and you can write a percent as a decimal, so 8% is 0.08. In this lesson you use those skills to rewrite expressions like $4(3n - 5) + 2n$ in a simpler way, without changing their value. The rewriting is the same arithmetic you already know, with a letter left in the answer.

3. Words in this lesson

4. Equivalent expressions: the same value in a different form

Two expressions are equivalent when they give the same number for every value of the letter. $2(x + 3)$ and $2x + 6$ are equivalent: when $x = 5$, both give 16, and when $x = -1$, both give 4. Rewriting an expression into an equivalent one never changes what it is worth; it only changes how it looks.

Three moves do almost all the work in grade 7:

1. Expand: multiply every term in a bracket by the number outside, $5(2k - 3) = 10k - 15$.
2. Collect like terms: add the coefficients of terms with the same letter, $10k - 15 + 4k = 14k - 15$.
3. Factor: take a common factor out of every term, $12k + 18 = 6(2k + 3)$.

All three come from the distributive property, $a(b + c) = ab + ac$, read forward to expand and backward to factor. Different forms show different things. A factored form shows a common part, and an expanded form shows the total at a glance.

5. Terms, coefficients and like terms

An expression such as $7x - 4 + 2x + 9$ is made of terms, and the sign in front of each term belongs to it: the terms are $7x$, $-4$, $2x$ and $9$. Like terms have the same letter part, so $7x$ and $2x$ can be combined, and so can $-4$ and 9. To combine like terms, add their coefficients: $7x + 2x = 9x$ and $-4 + 9 = 5$, so the expression simplifies to $9x + 5$.

Terms with different letter parts can never be combined. $9x + 5$ cannot become $14x$, because $9x$ counts groups of $x$ and 5 counts ones. It is like trying to add 9 bags of apples and 5 loose apples: you cannot give the total number of apples until you know how many are in a bag. Remember that $x$ by itself has a coefficient of 1, and $-x$ has a coefficient of $-1$.

6. Expanding: multiply every term inside

To expand $a(b + c)$, multiply the number outside by each term inside: $4(3n - 5) = 12n - 20$. The sign of each term goes along with it, so $4 \times (-5) = -20$.

When the number outside is negative, every sign inside is affected. $-3(2y - 7)$ means $-3 \times 2y$ plus $-3 \times (-7)$, which is $-6y + 21$. A fraction in front works the same way: $\tfrac{1}{2}(8m + 6) = 4m + 3$, because multiplying by one half is dividing by 2.

After expanding, look for like terms to collect. In $4(3n - 5) + 2n$ the expanded form is $12n - 20 + 2n$, and the two $n$ terms combine to give $14n - 20$.

7. Subtracting a bracket

A minus sign in front of a bracket means subtract everything inside. It is the same as multiplying the bracket by $-1$, so every sign inside flips:

$$-(3a - 5) = -3a + 5$$

So $(7a + 2) - (3a - 5) = 7a + 2 - 3a + 5 = 4a + 7$. The most common mistake is to flip only the first sign and write $7a + 2 - 3a - 5$. A quick way to catch it is to test a value. With $a = 1$, the original is $(7 + 2) - (3 - 5) = 9 - (-2) = 11$, and $4a + 7$ also gives 11, while the wrong version gives only 1.

8. Factoring: the distributive property backward

To factor $18w - 24$, find the greatest common factor of the terms. The largest number that divides both 18 and 24 is 6. Divide each term by 6 to see what stays inside the bracket: $18w \div 6 = 3w$ and $-24 \div 6 = -4$. So $18w - 24 = 6(3w - 4)$.

Always check by expanding: $6(3w - 4) = 18w - 24$. If you take out a smaller common factor, such as 2, you get $2(9w - 12)$. That is equivalent, but it is not fully factored, because 9 and 12 still share a factor of 3.

Sometimes it helps to take out a negative factor. $-10x + 15$ can be written as $-5(2x - 3)$; check it: $-5 \times 2x = -10x$ and $-5 \times (-3) = 15$.

9. Percent changes as one multiplier

A percent increase or decrease is a small linear expression. If a price $p$ goes up by 6%, the new price is the old price plus 6 hundredths of it: $p + 0.06p$. The term $p$ is $1 \cdot p$, so the like terms combine to $1.06p$.

A decrease works the same way with subtraction: 30% off leaves $p - 0.30p = 0.70p$. So one multiplication does the whole job. Take 30% off a 50-dollar pair of shoes by working out $0.70 \times 50 = 35$ dollars, without finding the 15-dollar discount first.

Two changes in a row multiply. A 10% increase followed by a 10% decrease is $0.90 \times 1.10p = 0.99p$, which is 1% less than you started with, not the same price.

10. Why rewriting is worth doing

Equivalent expressions give the same numbers, but each form tells you something different. $4(s + 2)$ says four people each paid $s + 2$ dollars. The expanded form $4s + 8$ says the drinks cost 8 dollars in all. $1.06p$ says the new price is 106% of the old one, while $p + 0.06p$ shows the increase on its own.

A simpler form is also easier to use. Working out $4(3n - 5) + 2n$ for $n = 7$ takes four operations, while $14n - 20$ needs only two. When you later solve equations and inequalities, the first job is often to rewrite each side in its simplest form.

11. The method, step by step, and how to check it

Use these steps to simplify a linear expression.

1. Deal with every bracket. Multiply each term inside by the number outside, sign included. A minus in front of a bracket flips every sign inside.
2. Write all the terms in one line, each with its own sign.
3. Collect like terms: add the coefficients of the letter terms, then add the plain numbers.
4. Write the simplest form, usually one letter term and one number, such as $14n - 20$.

To factor, find the greatest common factor, divide each term by it, and write it outside a bracket.

How to check. Choose an easy value for the letter, like 1 or 2, and work out the original expression and your answer. If the two numbers differ, something went wrong. Check a factored answer by expanding it back out. Avoid testing with 0 alone, because some mistakes only show when the letter is not zero.

12. In the world: fencing a dog run

A family wants a rectangular dog run whose length is 6 feet more than its width $w$. The fence goes around all four sides, so it needs $2w + 2(w + 6)$ feet. Expanding gives $2w + 2w + 12 = 4w + 12$ feet, and factoring gives $4(w + 3)$. The expanded form is easy to use: a run 8 feet wide needs $4(8) + 12 = 44$ feet of fence. The factored form says the same thing in another way: the fence is as long as 4 sides that are each $w + 3$ feet, which is a square that is 3 feet wider. If fencing costs 9 dollars a foot, the cost is $9(4w + 12) = 36w + 108$ dollars, or 396 dollars for the 8-foot run.

13. In the world: a sale price with sales tax

In Texas the combined state and local sales tax can be as high as 8.25%. Suppose a store takes 15% off a jacket that costs $p$ dollars, and then charges 8.25% tax. The sale price is $0.85p$, and adding tax multiplies by 1.0825, so the total is $1.0825 \times 0.85p \approx 0.92p$. One multiplier does everything: a jacket marked 80 dollars costs about $0.92 \times 80 = 73.60$ dollars at the register. Because multiplication can be done in either order, it does not matter whether the store applies the tax or the discount first.

14. In the world: tickets and snacks for a group

A group of $n$ friends goes to the movies. Each ticket costs 12.50 dollars and each friend buys a snack for 4.25 dollars. Adding the tickets and the snacks separately gives $12.50n + 4.25n$, and collecting like terms gives $16.75n$. For 6 friends, that is $16.75 \times 6 = 100.50$ dollars. The factored form $n(12.50 + 4.25)$ shows the same total as one person's cost multiplied by the size of the group.

15. Mistakes to avoid

The most common mistake is multiplying only the first term inside a bracket. $3(2x - 4)$ is $6x - 12$, not $6x - 4$.

The second is flipping only the first sign when a bracket is subtracted. $(7a + 2) - (3a - 5)$ is $4a + 7$, not $4a - 3$.

The third is combining terms that are not alike. $5x + 3$ stays as it is; it is not $8x$.

The fourth is losing the sign that belongs to a term. In $6 - 2x$, the term is $-2x$, so rearranging gives $-2x + 6$, not $2x + 6$.

The last is adding a percent as if it were dollars. A 15% increase on $p$ is $1.15p$, not $p + 15$.

16. Expand and simplify $4(3n - 5) + 2n$

1. Multiply the first term inside by 4.

   $4 \times 3n = 12n$

   The number outside multiplies every term inside the bracket.

2. Multiply the second term inside by 4, with its sign.

   $4 \times (-5) = -20$

   The minus sign belongs to the 5.

3. Write the expression without the bracket.

   $12n - 20 + 2n$

   The $+2n$ outside the bracket comes along unchanged.

4. Collect the $n$ terms.

   $12n + 2n = 14n, \quad \text{so } 14n - 20$

   They are like terms, so their coefficients add.

5. Check with $n = 2$.

   $4(6 - 5) + 4 = 8, \quad 14(2) - 20 = 8$

   Both forms give the same value, so they are equivalent.

17. Subtract a multiple of a bracket: $(5k - 3) - 2(k - 4)$

1. Treat the minus and the 2 together as a multiplier of $-2$.

   $(5k - 3) + (-2)(k - 4)$

   Subtracting 2 brackets is adding $-2$ brackets.

2. Multiply the $k$ term by $-2$.

   $-2 \times k = -2k$

   A negative times a positive is negative.

3. Multiply the $-4$ by $-2$.

   $-2 \times (-4) = 8$

   A negative times a negative is positive.

4. Write every term in one line.

   $5k - 3 - 2k + 8$

   The first bracket opens unchanged.

5. Collect the $k$ terms and the plain numbers.

   $5k - 2k = 3k, \quad -3 + 8 = 5$

   Like terms combine separately.

6. Check with $k = 1$.

   $(5 - 3) - 2(1 - 4) = 2 + 6 = 8, \quad 3(1) + 5 = 8$

   The answer $3k + 5$ agrees with the original.

18. A markup of 25%, then 20% off

1. Write the markup as an addition.

   $p + 0.25p$

   The new price is the old price plus 25 hundredths of it.

2. Collect the like terms.

   $p + 0.25p = 1.25p$

   The term $p$ is $1 \cdot p$.

3. Write 20% off as a multiplier.

   $q - 0.20q = 0.80q$

   Taking 20% away leaves 80% of the marked-up price $q$.

4. Apply the discount to the marked-up price.

   $0.80 \times 1.25p$

   The discount is taken from the new price, not the original.

5. Multiply the two decimals.

   $0.80 \times 1.25 = 1.00$

   Eight tenths of one and a quarter is exactly one.

6. Write the final price as an expression.

   $0.80 \times 1.25p = p$

   The final price equals the original price.

7. Check with a price of 40 dollars.

   $1.25 \times 40 = 50, \quad 0.80 \times 50 = 40$

   A real number agrees: the two changes cancel exactly.

19. Your turn: factor $15y - 35$

1. Find the greatest common factor of 15 and 35.

   $\gcd(15, 35) = 5$

   Five is the largest number that divides both.

2. Divide each term by 5.

   $15y \div 5 = 3y, \quad -35 \div 5 = -7$

   What is left goes inside the bracket.

3. Your turn: work this step out. Its working is at the end of the packet.

   Write the factored form and check it.

20. Guided practice

A price of $p$ dollars goes up by $23\%$. Which expression gives the new price?

• $1.23p$
• $0.23p$
• $p + 23$
• $23p$

21. Guided practice

Complete the worked solution to expand and simplify $-2(5x - 6) + 38x$.

1. Multiply the coefficient of the $x$ term inside by the number outside.

   $-2 \times 5 =$ _p

   A negative times a positive is negative.

2. Multiply the plain number inside by the number outside.

   $-2 \times (-6) =$ _q

   A negative times a negative is positive.

3. Add the two coefficients of $x$.

   $\text{the new coefficient} + 38 =$ _r

   The expanded $x$ term and the $38x$ are like terms; the plain number stays as it is.

22. Guided practice

Expand and simplify: $6(4x - 6) + x$.

Answer:

23. Practice

Simplify $(6m + 7) - (2m - 5)$.

Answer:

24. Practice

Factor $18y + 30$ by taking out the greatest common factor. Fill in the three blanks.

$18y + 30 = $ _g( _ay + _b )

25. Practice

Expand and simplify $\dfrac{1}{3}(24t - 21)$.

Answer:

26. Somewhere new

A rectangular garden is $w$ feet wide. Its length is $6$ feet less than $4$ times its width. Write a simplified expression for the length of fence needed to go all the way around it.

Answer:

27. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

28. Test question

Expand and simplify $8(x + 9) - 2(x - 3)$.

Answer:

29. What you can do now

You can rewrite linear expressions. Without looking: simplify $(6x + 1) - 3(x - 2)$, factor $14y - 21$, and write a 35% discount on a price $p$ as one multiplier.

Working for the steps left to you

19. Your turn: factor $15y - 35$, step 3