GED Math: Algebraic Expressions
The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both. Find the prime factorization of each, circle the primes they share, and multiply them. The GCF lets you factor expressions with the distributive property, as in 40 + 140 = 20(2 + 7). The least common multiple (LCM) is the smallest number both numbers divide into. Take each prime the greatest number of times it appears in either number, and multiply. The LCM is the best common denominator when you add or subtract fractions with large denominators. A shortcut check: GCF × LCM = the product of the two numbers.
1. Writing Algebraic Expressions
An algebraic expression uses numbers, operations, and variables (letters such as x and y that stand for unknown numbers) to show a relationship. Within one expression, the same letter always means the same number. Success on the GED means translating words into symbols. Look for key phrases: sum, increased by, more than mean add; difference, decreased by, less than mean subtract; product, times, of mean multiply; quotient, divided by, per mean divide; square means a power of 2. Watch the order: “6 less than a number” is x − 6, but “6 less a number” is 6 − x. Words like “the sum of” or “the difference of” followed by “and” tell you what goes inside parentheses.
Worked Example: Write “4 times the difference of −3 and a number” in symbols.
“The difference of −3 and a number” is (−3 − x), and “4 times” that whole quantity needs parentheses: 4(−3 − x).
| In words | In symbols |
|---|---|
| the product of 5 and a number | 5x |
| a number decreased by 12 | x − 12 |
| the sum of 3 and the square of a number | 3 + x2 |
| 6 less than the quotient of a number and 2 | x2 − 6 |
| one-half a number increased by 15 | 12x + 15 |
| 4 times the difference of −3 and a number | 4(−3 − x) |
| a number less another number | x − y |
| 10 less the square root of the sum of a number and 3 | 10 − √(x + 3) |
1. Which expression means “a number decreased by 9”?
Explanation: Decreased by means subtract the 9 from the number: x − 9.
2. Which expression means “7 more than the product of 4 and a number”?
Explanation: The product of 4 and a number is 4x. Seven more than that is 4x + 7.
3. Which expression means “5 less than the quotient of a number and 3”?
Explanation: The quotient of a number and 3 is x/3. Five less than that is x/3 − 5.
4. Which expression means “8 less than the square of a number”?
Explanation: Square the number first, then take 8 away: x2 − 8.
5. Which expression means “3 times the sum of a number and 6”?
Explanation: The sum of a number and 6 is (x + 6). Three times that sum is 3(x + 6).
6. Which expression means “the square root of the sum of a number and 4, subtracted from 10”?
Explanation: Subtracting something from 10 means 10 minus it, and the something is √(x + 4).
2. Setup Problems: Expressions from Real Situations
A setup problem describes a real situation and asks for the expression that models it. Pick out the quantity you want, name the unknowns, and build the expression piece by piece: a percent of one amount becomes a decimal times the variable, and a “per item” charge becomes the rate times the number of items. Use parentheses when one part has to be done before another. Test tip: check your choice by substituting easy numbers and asking whether the result makes sense for the situation.
Worked Example: The shipping on an order is 5% of the total cost plus $0.80 per item. Let c be the total cost in dollars and n the number of items. Find an expression for the shipping.
5% of the total is 0.05c, and $0.80 per item is 0.80n. Shipping = 0.05c + 0.80n. Check: for $100 and 2 items, 5 + 1.60 = $6.60, which is reasonable.
1. An order’s shipping is 5% of the total cost c plus $0.80 per item n. Which expression gives the shipping?
Explanation: The percent applies to the total cost c, and the per-item charge applies to the number of items n.
2. A club donates $2,000 plus $0.75 for each ticket sold over 1,500. If t tickets are sold and t is more than 1,500, which expression gives the donation?
Explanation: Only the tickets over 1,500 count, so subtract first: (t − 1500). Multiply by 0.75 and add to 2,000.
3. A store pays an hourly wage plus commission. Option B pays $8.00 per hour plus 2% of sales. If h is hours and s is sales in dollars, which expression gives weekly pay?
Explanation: Wage is $8 times hours (8h). Commission is 2% of sales (0.02s). Add the two.
4. The sum of 2 times a number and 5 times a second number is divided by the difference of 4 and a third number. Which expression shows this?
Explanation: Each “sum of” and “difference of” is its own group, so each gets parentheses.
5. A gym charges a $25 sign-up fee plus $12 per month. Which expression gives the cost for m months?
Explanation: The fee is paid once (25). The monthly cost grows with m (12m).
6. A plumber charges a $40 visit fee plus $35 per hour. Which expression gives the cost of a job that lasts h hours?
Explanation: $40 is charged once. $35 is charged for each of the h hours.
3. Simplifying: Combining Like Terms
A term is a number, a variable, or a product or quotient of numbers and variables. Terms are separated by plus and minus signs. Like terms have the same variable raised to the same power: 3x2 and 5x2 are like terms, and so are 8y and 4y, but 6x and 2x2 are not. To simplify an expression, combine like terms only (add or subtract the numbers in front of the variables, called coefficients). By custom, write the term with the greatest exponent first and continue in descending order. Remember that 1x means x.
Worked Example: Simplify 2x − 5 + 4x2 − 8 + 6x.
Group like terms: (2x + 6x) + (−5 − 8) + 4x2 = 8x − 13 + 4x2. In descending order: 4x2 + 8x − 13.
1. Simplify 5x + 3 + 2x − 7.
Explanation: 5x + 2x = 7x and 3 − 7 = −4.
2. Which term is like 6x?
Explanation: Like terms have the same variable to the same power. 9x matches 6x.
3. Simplify 4y + 9 + 11y + 6.
Explanation: 4y + 11y = 15y and 9 + 6 = 15.
4. Simplify 3x2 + 4x − x2 + 9.
Explanation: 3x2 − x2 = 2x2. The 4x and the 9 have no like terms.
5. Simplify 5y2 + 4 − 3y2 + 5 + y.
Explanation: 5y2 − 3y2 = 2y2; the y stays; 4 + 5 = 9.
6. Simplify x2 + 7x − 2x2 − x.
Explanation: x2 − 2x2 = −x2, and 7x − x = 6x.
4. The Distributive Property
The distributive property removes grouping symbols: a(b + c) = ab + ac and a(b − c) = ab − ac. Multiply the term outside by every term inside, then combine like terms. If a minus sign sits in front of parentheses, imagine a 1 there: −(2x + 3) = −1(2x + 3) = −2x − 3. Every sign inside flips when you distribute a negative.
Worked Example: Simplify 6x − 4(x + 5) + 8.
Distribute −4: 6x − 4x − 20 + 8. Combine: 2x − 12. Answer: 2x − 12.
1. Simplify 3(x + 5).
Explanation: Multiply 3 by both terms: 3x + 15.
2. Simplify −(3x − 5).
Explanation: Distribute −1: −3x and −(−5) = +5.
3. Simplify 4x − 2(x − 6).
Explanation: −2(x − 6) = −2x + 12. Then 4x − 2x + 12 = 2x + 12.
4. Simplify 5(y + 2) + 3(y − 4).
Explanation: 5y + 10 + 3y − 12 = 8y − 2.
5. Simplify 6x − 4(x + 5) + 8.
Explanation: 6x − 4x − 20 + 8 = 2x − 12.
6. Simplify −2(x + 3) − 3(x + 1).
Explanation: −2x − 6 − 3x − 3 = −5x − 9.
7. Simplify 7 − (x − 3) + 2x.
Explanation: 7 − x + 3 + 2x = x + 10.
5. Evaluating Expressions
To evaluate an expression, substitute the given number for each variable (put the number in parentheses, especially a negative one) and follow the order of operations. With a fraction bar, simplify the top and bottom first and then divide. Evaluating is also how you use formulas, such as the temperature conversions C = 59(F − 32) and F = 95C + 32.
Worked Example: Find the value of 3x + 2y4 when x = 6 and y = 5.
Substitute: 3(6) + 2(5)4. Top: 18 + 10 = 28. Divide: 28 ÷ 4 = 7.
1. Find 5(x + 2) + 7 when x = 3.
Explanation: 5(5) + 7 = 25 + 7 = 32.
2. Find 2x2 + 3(x − 1) when x = 4.
Explanation: 2(16) + 3(3) = 32 + 9 = 41.
3. Find 5x + y3 when x = 4 and y = 1.
Explanation: Top: 20 + 1 = 21. Divide by 3: 7.
4. Find y2 + 10 − (y − 4)2 when y = 3.
Explanation: 9 + 10 − (−1)2 = 19 − 1 = 18.
5. Find (x + y)22 − 10 when x = 2 and y = 4.
Explanation: (6)2 = 36, divided by 2 is 18. Then 18 − 10 = 8.
6. Find 8x + 9y − (2x + y) when x = 4 and y = 6.
Explanation: Simplify: 6x + 8y = 24 + 48 = 72.
7. Find x2 + 3x − 4 when x = −3.
Explanation: (−3)2 = 9 and 3(−3) = −9. 9 − 9 − 4 = −4.
8. Use F = 95C + 32. What is F when C = 40?
Explanation: 9/5 × 40 = 72, and 72 + 32 = 104.
9. Use C = 59(F − 32). What is C when F = 95?
Explanation: 95 − 32 = 63, and 5/9 × 63 = 35.
10. Given 2x2 − 3(y + 2), which values give 9?
Explanation: For x = 3, y = 1: 18 − 9 = 9. The others give −1, −7, and −10.
6. Factors, Multiples, and Prime Factorization
When you multiply two numbers to get a product, the numbers you multiplied are factors of that product. A T-chart lists every factor pair: write 1 and the number, then test 2, 3, 4, and so on, stopping when the columns meet. Every positive factor also has a negative partner (for example, −3 × −8 = 24), but test questions usually ask for positive factors. Prime numbers are whole numbers greater than 1 with only two positive factors, 1 and themselves (2, 3, 5, 7, 11, 13, …). The number 1 is not prime. The prime factorization of a number is the product of primes that equals it; build it with a factor tree, splitting any composite number until every branch ends in a prime.
Worked Example: What are all the positive factors of 24?
Build the T-chart: 1 × 24, 2 × 12, 3 × 8, 4 × 6. The factors are 1, 2, 3, 4, 6, 8, 12, 24. Its prime factorization is 24 = 2 × 2 × 2 × 3.
1. Which list shows all the positive factors of 18?
Explanation: Pairs: 1 × 18, 2 × 9, 3 × 6. That gives 1, 2, 3, 6, 9, 18.
2. How many positive factors does 20 have?
Explanation: Pairs: 1 × 20, 2 × 10, 4 × 5. That is 6 factors.
3. Which number is prime?
Explanation: 29 has only 1 and 29 as factors. 27 = 3 × 9, 51 = 3 × 17, and 1 is not prime.
4. What is the prime factorization of 36?
Explanation: 36 = 6 × 6 = (2 × 3) × (2 × 3). Only the first choice uses all primes.
5. What is the prime factorization of 54?
Explanation: 54 = 2 × 27 = 2 × 3 × 9 = 2 × 3 × 3 × 3.
6. What is the prime factorization of 70?
Explanation: 70 = 7 × 10 = 7 × 2 × 5. The other choices include a non-prime.
7. Which of these is a factor of 56?
Explanation: 56 ÷ 14 = 4, with no remainder.
7. Greatest Common Factor and Least Common Multiple
The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both. Find the prime factorization of each, circle the primes they share, and multiply them. The GCF lets you factor expressions with the distributive property, as in 40 + 140 = 20(2 + 7). The least common multiple (LCM) is the smallest number both numbers divide into. Take each prime the greatest number of times it appears in either number, and multiply. The LCM is the best common denominator when you add or subtract fractions with large denominators. A shortcut check: GCF × LCM = the product of the two numbers.
Worked Example: Find the GCF and LCM of 36 and 60.
36 = 2 × 2 × 3 × 3 and 60 = 2 × 2 × 3 × 5. Shared primes: 2, 2, 3, so GCF = 12. Take 2 twice, 3 twice, and 5 once: LCM = 2 × 2 × 3 × 3 × 5 = 180. Check: 12 × 180 = 2,160 = 36 × 60.
1. What is the GCF of 36 and 60?
Explanation: The shared primes are 2 × 2 × 3 = 12.
2. What is the LCM of 36 and 60?
Explanation: Use 2 × 2 × 3 × 3 × 5 = 180.
3. What are the GCF and LCM of 18 and 27?
Explanation: 18 = 2 × 3 × 3 and 27 = 3 × 3 × 3. GCF = 3 × 3 = 9. LCM = 2 × 3 × 3 × 3 = 54.
4. What are the GCF and LCM of 50 and 125?
Explanation: 50 = 2 × 5 × 5 and 125 = 5 × 5 × 5. GCF = 25. LCM = 2 × 5 × 5 × 5 = 250.
5. Which expression equals 24 + 60?
Explanation: The GCF is 12: 24 = 12 × 2 and 60 = 12 × 5.
6. Find 712 + 518.
Explanation: LCM of 12 and 18 is 36. 21/36 + 10/36 = 31/36.
7. Find 730 − 245.
Explanation: LCM of 30 and 45 is 90. 21/90 − 4/90 = 17/90.
8. One bus leaves a station every 12 minutes and another every 20 minutes. They leave together at 8:00 a.m. In how many minutes do they next leave together?
Explanation: This is the LCM of 12 and 20: 2 × 2 × 3 × 5 = 60.
Common Mistakes to Avoid
- Combining unlike terms, such as 6x + 2x2. Only terms with the same variable and power combine.
- Distributing to only the first term: 3(x + 5) is 3x + 15, not 3x + 5.
- Not flipping signs inside parentheses after a minus: −(x − 3) = −x + 3.
- Mixing up “less than” (order reverses) with “less”: 6 less than x is x − 6.
- Forgetting parentheses around sums and differences that are multiplied or divided.
- Skipping parentheses when substituting a negative number: (−3)2 = 9, not −9.
- Treating 1 as a prime number. Prime numbers start at 2.
- Mixing up GCF and LCM: the GCF is never larger than either number; the LCM is never smaller.
Frequently Asked Questions
What is an algebraic expression?
A combination of numbers, operations, and variables, with no equals sign, that shows a number relationship.
What are like terms?
Terms with the same variable raised to the same power, such as 8y and 4y. Only like terms can be combined.
How do I use the distributive property?
Multiply the term outside the parentheses by every term inside, then combine like terms: a(b + c) = ab + ac.
How do I evaluate an expression?
Replace each variable with its value, using parentheses, and follow the order of operations.
How do I find the GCF and LCM?
Write the prime factorization of each number. The GCF multiplies the shared primes. The LCM multiplies each prime the greatest number of times it appears.
Why is 1 not a prime number?
A prime number has exactly two positive factors, 1 and itself. The number 1 has only one factor.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.