Linear expressions: combine, factor, expand

1. List each term with the sign in front of it: +5x, +3, -2x, +7.
2. Group the like terms: (5x - 2x) + (3 + 7).
3. Combine the x-terms: 5x - 2x = 3x.
4. Combine the constants: 3 + 7 = 10.

The simplified expression is 3x + 10.

Check: let x = 2. Original: 5(2) + 3 - 2(2) + 7 = 10 + 3 - 4 + 7 = 16. Answer: 3(2) + 10 = 16 ✓

Tip: the minus sign belongs only to the term right after it (-2x), not to the terms that come later.

1. Factors of 6: 1, 2, 3, 6. Factors of 15: 1, 3, 5, 15.
2. The greatest common factor is 3.
3. Divide each term by 3: 6x \div 3 = 2x and 15 \div 3 = 5.
4. Write the GCF outside: 6x + 15 = 3(2x + 5).

The answer is 3(2x + 5).

Check: expand: 3 \cdot 2x + 3 \cdot 5 = 6x + 15 ✓. The terms inside, 2x and 5, have no common factor left, so it is factored completely.

Tip: you can only pull out x when every term contains x. Here 15 has no x, so the GCF is just the number 3.

1. Write every term with its own sign: +7y, -4, -9y, +11.
2. Group like terms: (7y - 9y) + (-4 + 11).
3. Combine the y-terms: 7y - 9y = -2y (you owe 9 and have 7, so you are 2 short).
4. Combine the constants: -4 + 11 = 7.

The simplified expression is -2y + 7.

Check: let y = 1. Original: 7 - 4 - 9 + 11 = 5. Answer: -2(1) + 7 = 5 ✓

Tip: a negative result for a coefficient is fine. Do not "fix" -2y into 2y - the sign is part of the answer.

1. Factors of 14: 1, 2, 7, 14. Factors of 35: 1, 5, 7, 35.
2. The greatest common factor is 7.
3. Divide each term by 7: 14n \div 7 = 2n and 35 \div 7 = 5.
4. Write the GCF outside the parentheses: 14n - 35 = 7(2n - 5).

The answer is 7(2n - 5).

Check: expand: 7 \cdot 2n - 7 \cdot 5 = 14n - 35 ✓. Inside, 2n and 5 share no factor other than 1, so the factoring is complete.

Tip: after factoring, always look inside the parentheses. If the terms there still share a factor, you did not use the greatest one.

1. Distribute in the second part: 15(x + 3) = 15x + 45.
2. Write the whole expression: 15x + 15x + 45.
3. Combine the x-terms: 15x + 15x = 30x.
4. The constant stays: 45.

His pay is 30x + 45 dollars.

Check: let x = 4. Saturday 4 h, Sunday 7 h, total 11 h, and 11 \times 15 = 165. Answer: 30(4) + 45 = 165 ✓

Tip: the 45 has a meaning: the 3 extra Sunday hours are worth 3 \times 15 = 45 dollars. Thinking about meaning is a fast way to check a simplified expression.

1. Distribute 3: 3(2x - 4) = 6x - 12.
2. Distribute -2: (-2)(x) = -2x and (-2)(-5) = +10, so -2(x - 5) = -2x + 10.
3. Put it together: 6x - 12 - 2x + 10.
4. Combine the x-terms: 6x - 2x = 4x.
5. Combine the constants: -12 + 10 = -2.

The answer is 4x - 2.

Check: let x = 2. Original: 3(0) - 2(-3) = 0 + 6 = 6. Answer: 4(2) - 2 = 6 ✓

Tip: treat a minus sign in front of parentheses as part of the number you distribute. Two negatives multiply to a positive.

1. Distribute 8: 8(50 - x) = 8 \cdot 50 - 8 \cdot x = 400 - 8x.
2. Write the whole expression: 12x + 400 - 8x.
3. Combine the x-terms: 12x - 8x = 4x.
4. The constant is 400.

Total sales: 4x + 400 dollars.

Check: if all 50 were adult tickets (x = 50): 12(50) + 8(0) = 600, and 4(50) + 400 = 600 ✓. If none were adult tickets (x = 0): 8(50) = 400 and 4(0) + 400 = 400 ✓

Tip: the simplified form has meaning. $400 is what 50 child tickets would bring in, and each adult ticket adds $4 more because it costs $4 more than a child ticket.

1. Distribute \tfrac{1}{2}: \tfrac{1}{2} \cdot 8x = 4x and \tfrac{1}{2} \cdot 6 = 3, so the first part is 4x + 3.
2. Distribute the negative: -(x - 4) = -x + 4.
3. Combine: 4x + 3 - x + 4.
4. x-terms: 4x - x = 3x. Constants: 3 + 4 = 7.

The answer is 3x + 7.

Check: let x = 2. Original: \tfrac{1}{2}(16 + 6) - (2 - 4) = 11 - (-2) = 13. Answer: 3(2) + 7 = 13 ✓

Tip: a fraction outside the parentheses distributes exactly like a whole number. Halving each term is often easiest to do in your head.

1. Distribute 5: 5(x - 2) = 5x - 10. Tyrell got this part right.
2. Distribute -3: (-3)(x) = -3x and (-3)(-4) = +12. So -3(x - 4) = -3x + 12.
3. Tyrell wrote -12 instead of +12, so his Step 1 is wrong.
4. Correct work: 5x - 10 - 3x + 12 = 2x + 2.

The true statement is: the error is in Step 1; it should end in +12, and the correct answer is 2x + 2.

Check: let x = 5. Original: 5(3) - 3(1) = 15 - 3 = 12. Correct answer: 2(5) + 2 = 12 ✓. Tyrell's: 2(5) - 22 = -12 ✗

Tip: when a minus sign sits in front of a number and parentheses, distribute the negative number. A negative times a negative is positive.

1. Left rectangle: 4 \cdot 3x = 12x.
2. Right rectangle: 4 \cdot 7 = 28.
3. Total area: 12x + 28.
4. This matches the distributive property: the large rectangle is 4 by (3x + 7), and 4(3x + 7) = 12x + 28.

The total area is 12x + 28.

Check: let x = 2. The large rectangle is 4 by 3(2) + 7 = 13, so the area is 52. Answer: 12(2) + 28 = 52 ✓

Tip: an area model is a picture of the distributive property. Each part of the rectangle is one product, and the total is the sum of the parts.

1. Distribute \tfrac{2}{3}: \tfrac{2}{3} \cdot 9x = 6x and \tfrac{2}{3} \cdot 12 = 8. So \tfrac{2}{3}(9x - 12) = 6x - 8.
2. Distribute \tfrac{1}{4}: \tfrac{1}{4} \cdot 8x = 2x and \tfrac{1}{4} \cdot 4 = 1. So \tfrac{1}{4}(8x + 4) = 2x + 1.
3. Add: 6x - 8 + 2x + 1.
4. Combine: 6x + 2x = 8x and -8 + 1 = -7.

The answer is 8x - 7.

Check: let x = 3. Original: \tfrac{2}{3}(15) + \tfrac{1}{4}(28) = 10 + 7 = 17. Answer: 8(3) - 7 = 17 ✓

Tip: to multiply a whole number by a fraction like \tfrac{2}{3}, divide by the bottom (3) first, then multiply by the top (2): 12 \div 3 \times 2 = 8. Smaller numbers, fewer mistakes.

1. 4(2x - 3) = 8x - 12.
2. -3(x - 2) = -3x + 6 because (-3)(-2) = +6.
3. +2(5 - x) = 10 - 2x.
4. Write all terms: 8x - 12 - 3x + 6 + 10 - 2x.
5. x-terms: 8x - 3x - 2x = 3x.
6. Constants: -12 + 6 + 10 = 4.

The answer is 3x + 4.

Check: let x = 1. Original: 4(-1) - 3(-1) + 2(4) = -4 + 3 + 8 = 7. Answer: 3 + 4 = 7 ✓

Tip: with three or more groups, write every term out on one line before combining. Then cross off each term as you add it, so none get lost.

1. 2(3x - 5) = 6x - 10.
2. -4(x - 1) = -4x + 4 (because (-4)(-1) = +4).
3. 3(x + 2) = 3x + 6.
4. All terms: 6x - 10 - 4x + 4 + 3x + 6.
5. x-terms: 6x - 4x + 3x = 5x. Constants: -10 + 4 + 6 = 0.

The expression simplifies to 5x.

Check: let x = 2. Original: 2(1) - 4(1) + 3(4) = 2 - 4 + 12 = 10. Answer: 5(2) = 10 ✓

Tip: a constant of 0 is a real result, not a mistake. Write 5x, not 5x + 0.

1. Expand: a(2x + 3) - 4x = 2ax + 3a - 4x = (2a - 4)x + 3a.
2. Match the constants: 3a = 15, so a = 5.
3. Check with the x-coefficients: 2a - 4 = 2(5) - 4 = 6 ✓ (matches 6x).

The value is a = 5.

Check: 5(2x + 3) - 4x = 10x + 15 - 4x = 6x + 15 ✓

Tip: when an unknown multiplies a group, the constant term often gives the quickest equation. Use the other term as your check.

1. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 45: 1, 3, 5, 9, 15, 45. The GCF is 9.
2. Ken: 3(6x - 15) = 18x - 45. Equivalent, but 6x and 15 still share a 3, so it is not factored completely.
3. Lia: 9(2x - 5) = 18x - 45. Equivalent, and 2x and 5 share no factor. Factored completely.
4. Max: 9(2x - 45) = 18x - 405. Not equivalent; he divided only the first term.

The true statement is only Lia's answer is factored completely.

Check: with x = 3: 18(3) - 45 = 9. Ken: 3(3) = 9 ✓, Lia: 9(1) = 9 ✓, Max: 9(-39) = -351 ✗

Tip: "factored completely" means two things: the factored form expands back to the original, and nothing common is left inside the parentheses.

1. Room A: 4(x + 2) = 4x + 8.
2. Room B: 6(x - 1) = 6x - 6.
3. Total: 4x + 8 + 6x - 6.
4. Combine: 4x + 6x = 10x and 8 - 6 = 2.

The total floor area is 10x + 2 square feet.

Check: let x = 10. Room A is 4 \times 12 = 48, Room B is 6 \times 9 = 54, total 102. Answer: 10(10) + 2 = 102 ✓

Tip: put parentheses around any side length that has two terms before you multiply. 4 \cdot x + 2 is not the same as 4(x + 2).

1. Inside the bracket: 4(n + 25) = 4n + 100, so the costs are 12n + 4n + 100 = 16n + 100.
2. Subtract the whole bracket: 45n - (16n + 100) = 45n - 16n - 100.
3. Combine: 45n - 16n = 29n.

The profit is 29n - 100 dollars.

Check: let n = 10. Money in: 450. Costs: 120 + 4(35) = 260. Profit: 450 - 260 = 190. Answer: 29(10) - 100 = 190 ✓

Tip: subtracting a bracket subtracts every cost inside it. The minus sign in front changes +100 to -100.

1. Correct work: distribute -3: (-3)(2x) = -6x and (-3)(-6) = +18.
2. So 4 - 3(2x - 6) + 5x = 4 - 6x + 18 + 5x.
3. Combine: -6x + 5x = -x and 4 + 18 = 22. The correct result is -x + 22.
4. Dana wrote -18 instead of +18: a sign error in Step 1.
5. Ben subtracted 4 - 3 = 1 before multiplying: an order-of-operations error in Step 1.

The true statement is both students made an error; the correct answer is -x + 22.

Check: let x = 2. Original: 4 - 3(-2) + 10 = 4 + 6 + 10 = 20. Correct answer: -2 + 22 = 20 ✓. Dana: -16 ✗. Ben: 8 ✗

Tip: in error-analysis questions, work the problem yourself before reading the students' work, so their mistakes do not steer you.