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Lesson 5 of 11

Expressions using Letter-Numbers · Lesson 5 of 11

Adding and Subtracting Algebraic Expressions

“Combine whole expressions carefully, especially when a minus sign applies to a bracket.”

Learning Objectives

• Add expressions by grouping like signed terms. • Subtract an entire expression by changing every term’s sign. • Expand a multiple of a bracket using the distributive property. • Model net payments and totals across several rounds. • Interpret “subtract A from B” correctly and check results by substitution.

Combining Whole Expressions

An expression can describe one day’s income, one round’s points, or one part of a calculation. To find a combined total, add the complete expressions and then collect like terms. Brackets show where each original expression begins and ends; they help preserve its structure while you combine it with another.

For example, adding 4d − 7c + 9 and 8c − 11 + 9d gives (4d − 7c + 9) + (8c − 11 + 9d). A plus before a bracket adds the bracket exactly as written, so its internal signs stay unchanged. Then the d terms, c terms, and constant terms can be grouped separately.

Example — Adding expressions with constants

Problem
Add 4d − 7c + 9 and 8c − 11 + 9d.

  1. 1.Remove brackets after a plus: 4d − 7c + 9 + 8c − 11 + 9d.
  2. 2.Combine d terms: 4d + 9d = 13d. Combine c terms: −7c + 8c = c.
  3. 3.Combine numbers: 9 − 11 = −2. The sum is 13d + c − 2.

A Refund Is Subtracted as a Whole

A furniture shop charges ₹40 per chair and ₹75 per table. When the furniture is returned, it refunds ₹6 per chair and ₹10 per table. If x chairs and y tables are rented, the initial payment and the refund are each whole expressions. The net payment is the first amount minus the entire refund.

ItemInitial charge per itemRefund per itemNet cost per item
Chair₹40₹6₹34
Table₹75₹10₹65

The refund is 6x + 10y rupees, so subtracting it means taking away both 6x and 10y. If you subtract only the first term, you would wrongly add the table refund to the amount kept by the shop. Thinking about the money gives a reason for the sign changes.

Example — Net rental payment

Problem
Simplify (40x + 75y) − (6x + 10y). Find its value for 3 chairs and 2 tables.

  1. 1.Remove the refund bracket with a minus on each term: 40x + 75y − 6x − 10y.
  2. 2.Collect like terms: (40 − 6)x + (75 − 10)y = 34x + 65y.
  3. 3.For x = 3 and y = 2, net payment is 34 × 3 + 65 × 2 = 102 + 130 = ₹232.
  4. 4.Check separately: the initial charge is ₹270 and the refund is ₹38, giving ₹232.
Subtract a complete expressionLaTeX
Subtracting a bracket changes the sign of every signed term inside it.

You may also write the rental expression as (40x + 75y) + (−6x − 10y). This is addition of the negative of the refund. It gives exactly the same result because subtracting a quantity and adding its negative are the same operation.

Totals and Differences in a Points Game

Suppose each successful answer earns p points and each unsuccessful answer loses q points. A round with seven successes and three unsuccessful answers gives 7p − 3q points. The number of successes and failures supplies the coefficients, while p and q represent the size of each reward and penalty.

Example — Three rounds together

Problem
The round totals are 7p − 3q, 8p − 4q, and 6p − 2q. Find their combined total and its value for p = 4 and q = 1.

  1. 1.Add all three expressions: (7p − 3q) + (8p − 4q) + (6p − 2q).
  2. 2.The p terms give (7 + 8 + 6)p = 21p. The q terms give (−3 − 4 − 2)q = −9q.
  3. 3.The total is 21p − 9q. For p = 4 and q = 1, it is 84 − 9 = 75 points.
  4. 4.The separate rounds give 25, 28, and 22 points; their sum is also 75. With no penalty, q = 0 and the total becomes 21p.
Example — Comparing two totals

Problem
Another player has total 23p − 7q. How much greater is it than 21p − 9q?

  1. 1.Subtract the first player’s whole total: (23p − 7q) − (21p − 9q).
  2. 2.Remove the second bracket: 23p − 7q − 21p + 9q.
  3. 3.Collect terms: 2p + 2q. For p = 4, q = 1 the difference is 10 points.
  4. 4.When p > 0 and q ≥ 0, this difference is positive. The comparison relies on these reward-and-penalty meanings; arbitrary letter values would not always give a positive difference.

A total does not determine a unique collection of rounds. For example, 8p − 2q, 9p − 3q, and 6p − 2q add to 23p − 7q. Other round expressions can produce the same overall total, as long as their signed coefficients add to 23 and −7.

Expand Before Collecting

When a number multiplies a bracket, it multiplies every term inside. Expanding reveals which terms can be collected with terms outside the bracket. The purpose is to expose the full sum without changing its value.

Example — A multiple of a sum

Problem
Simplify 4(x + y) − y.

  1. 1.Distribute 4 across the bracket: 4x + 4y − y.
  2. 2.The x term stays 4x. The y terms combine as (4 − 1)y = 3y.
  3. 3.The result is 4x + 3y. At x = 2, y = 5, both forms give 23.
Example — Subtract A from B

Problem
Subtract 9a − 6b + 14 from 6a + 9b − 18.

  1. 1.“Subtract A from B” means B − A. Write (6a + 9b − 18) − (9a − 6b + 14).
  2. 2.Change every sign in the expression being removed: 6a + 9b − 18 − 9a + 6b − 14.
  3. 3.Combine: (6 − 9)a + (9 + 6)b + (−18 − 14) = −3a + 15b − 32.

A useful check is to choose simple numerical values and compare the unsimplified and simplified expressions. Such a check can catch a sign error. The reason that a result holds generally, however, is the arithmetic rule used at each step: distribution, sign changes, and collection of like terms.

Common mistake

The minus in B − (A) acts on every term of A, even a term that was already negative. Also, “subtract A from B” starts from B. Reversing the order usually changes the answer to its negative.

Check Your Understanding

Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.

Quiz

Quick check

What is (5x + 2y) + (3x − y)?

Quick check

What is (40x + 75y) − (6x + 10y)?

Quick check

Simplify 4(x + y) − y.

Quick check

Which expression means “subtract A from B”?

Quick check

In a game where q is the penalty per unsuccessful answer, what does no penalty mean?

Keep internal signs after addition, then collect 5x + 3x and 2y − y.

Practice Problems

Practice Problems
  1. Simplify p + q − (p + q), 2d − d − (d − c), and 2d − (d − d) − c.
  2. Give one possible set of three round expressions that total 23p − 7q.
  3. Add −6f + 19 − 8s and −23 + 13f + 12s.
  4. Add 8d − 14c + 9 and 16c − (11 + 9d).
  5. Add 6f − 20 + 8s and 23 − 13f − 12s.
  6. Add 13m − 12n and 12n − 13m. Then add −26m + 24n and 26m − 24n.
  7. Subtract −15x + 13 − 9y from 7y − 10 + 3x.
  8. Subtract 17g + 9 − 7h from 11 − 10g + 3h.
  9. Subtract 9a − 6b + 14 from 6a − (9b + 18).
  10. Subtract 10x + 2 + 10y from −3y + 8 − 3x.
  11. Subtract 8g + 4h − 10 from 7h − 8g + 20.
  12. Find the net cost of 10 chairs and 4 tables under the rental terms in this lesson.

0; c; 2d − c. Opening a subtracted bracket changes all its signs.

Key Takeaways

Key Takeaways

• Add whole expressions before collecting like terms. • A plus before a bracket preserves the internal signs. • A minus before a bracket changes every internal sign. • Distribute a multiplier to every term in its bracket. • “Subtract A from B” means B − A. • Interpret letters in context before making claims about which total is greater.

Previous · Lesson 4

Simplification of Algebraic Expressions

Next · Lesson 6

Equivalent Expressions and Correcting Mistakes