Expressions using Letter-Numbers · Lesson 3 of 11
Revisiting Arithmetic Expressions and Evaluating Algebraic Expressions
“Read signed terms correctly and substitute numbers without losing operation order.”
• Evaluate arithmetic expressions using brackets and multiplication correctly. • Rewrite subtraction as addition of a signed term. • Swap and group complete signed terms without changing their value. • Read multiplication written without a multiplication sign. • Evaluate expressions with positive and negative substitutions and explain common errors.
Arithmetic Rules Still Apply
Letters do not introduce a new set of arithmetic rules. Once their values are supplied, an algebraic expression becomes an ordinary number calculation. A secure understanding of brackets, multiplication, addition, and subtraction is therefore the starting point for evaluating expressions with letters.
In 23 − 10 × 2, the product 10 × 2 must be found before subtracting. Subtracting 10 first would change the meaning of the expression. Brackets can deliberately group a calculation to be done first, as in 8 × (16 − 6). Read what each operation acts on before beginning.
Problem
Evaluate 23 − 10 × 2 and 20 + 8 × (16 − 6).
- 1.For the first expression, compute the product: 10 × 2 = 20.
- 2.Then subtract: 23 − 20 = 3.
- 3.For the second, evaluate the bracket: 16 − 6 = 10. Multiply to get 8 × 10 = 80, then add 20 to get 100.
A part of an expression viewed as a sum, together with its positive or negative sign. For instance, 83 + 28 − 13 + 32 has signed terms 83, 28, −13, and 32.
Writing subtraction as addition of a negative term helps us regroup safely. The expression 83 + 28 − 13 + 32 is 83 + 28 + (−13) + 32. We may swap the complete signed terms and group 83 with −13, then 28 with 32. We may not move 13 and leave its negative sign behind.
Problem
Evaluate 83 + 28 − 13 + 32 by grouping signed terms.
- 1.Identify the four terms: 83, 28, −13, and 32.
- 2.Group 83 + (−13) = 70 and 28 + 32 = 60.
- 3.Add 70 + 60 = 130. The regrouping is allowed because it rearranges a sum of complete signed terms.
Brackets with a Negative Sign Outside
Subtracting a bracket means subtracting its whole value. For 68 − (18 + 13), first adding within the bracket gives 68 − 31. Another route is to remove the bracket by subtracting each contribution. Both routes agree because the same total is being removed.
If the bracket itself contains subtraction, the second sign also changes. For example, 10 − (6 − 2) = 10 − 4 = 6. Removing the bracket gives 10 − 6 + 2, again 6. We will use this rule with letters, but checking a small numerical example helps explain why a minus cannot simply be ignored.
Problem
Evaluate 68 − (18 + 13) and 42 + 15 − (8 − 7).
- 1.For the first, 18 + 13 = 31, then 68 − 31 = 37. Alternatively, 68 − 18 − 13 = 50 − 13 = 37.
- 2.For the second, 8 − 7 = 1, so 42 + 15 − 1 = 56.
- 3.Opening the second bracket gives 42 + 15 − 8 + 7 = 56. The last sign becomes plus because the entire difference was subtracted.
Omission of the Multiplication Symbol in Algebraic Expressions
Algebra often shortens a product by placing a number immediately before a letter. Thus 4n means 4 × n, and 7k means 7 × k. This convention saves space, but it does not turn a letter into a digit or change multiplication into addition.
Consider the sequence 4, 8, 12, 16, and so on. Its third entry is 4 × 3 and its twenty-ninth entry is 4 × 29. If n represents the position, counting from 1, the entry at that position is 4n. The letter n counts the position; the value 4n is the number found there.
Problem
Find 7k when k = 4, find 5m + 3 when m = 2, and find the twenty-ninth multiple of 4.
- 1.Read 7k as 7 × k. Replacing k by 4 gives 7 × 4 = 28.
- 2.Read 5m + 3 as 5 × m + 3. Replacing m by 2 gives 10 + 3 = 13.
- 3.The sequence formula is 4n. At n = 29 it gives 4 × 29 = 116.
If d = 6, then 3d = 18, not 36. Algebraic juxtaposition means multiplication: 3 × 6. The numeral 36 uses place value to mean thirty-six, which is a different notation.
Substitution, Including Negative Numbers
Substitution means replacing each occurrence of a letter by its assigned value. Put a negative value inside brackets while writing the calculation so that its sign remains visible. Then evaluate the expression using the arithmetic rules already discussed.
Problem
Evaluate 10 − a for a = −4 and 3(m + 1) for m = −6.
- 1.Replacing a by −4 gives 10 − (−4). Subtracting a negative adds its opposite, so the value is 14.
- 2.For the second expression, replace m inside the bracket: 3(−6 + 1).
- 3.The bracket is −5. The product 3 × (−5) is −15. Multiplication acts on the whole bracket.
For several letters, replace each one according to its own definition. If f = 3 and g = 1, then 2f − 2g becomes 2 × 3 − 2 × 1 = 4. The letters disappear after substitution, but their positions and the operations connecting them stay exactly where they were.
When checking an evaluation, first write the complete substitution. Next locate the first incorrect operation. Naming that error teaches more than replacing the final answer alone.
| Given values | Expression | Correct evaluation |
|---|---|---|
| a = −4 | 10 − a | 10 − (−4) = 14 |
| d = 6 | 3d | 3 × 6 = 18 |
| s = 7 | 3s − 2 | 21 − 2 = 19 |
| r = 8 | 2r + 1 | 16 + 1 = 17 |
| j = 5 | 2j | 2 × 5 = 10 |
| m = −6 | 3(m + 1) | 3 × (−5) = −15 |
| f = 3, g = 1 | 2f − 2g | 6 − 2 = 4 |
| t = 4, b = 3 | 2t + b | 8 + 3 = 11 |
| h = 5, n = 6 | h − (3 − n) | 5 − (−3) = 8 |
The evaluation 2j = 10 for j = 5 is already correct. Error-detecting work should include the possibility that no error exists. A habit of explaining each operation lets you accept a correct result and diagnose an incorrect one for the same reason.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
What is 34 − 14 + 20?
If d = 6, what does 3d equal?
If a = −4, what is 10 − a?
If m = −6, what is 3(m + 1)?
When rearranging 83 + 28 − 13 + 32 as a sum, which item must stay together?
Subtract 14 from 34 to get 20, then add 20 to get 40.
Practice Problems
- Evaluate 34 − 14 + 20, 7 × 4 + 9 × 6, and 20 + 8 × (16 − 6).
- Find the third and twenty-ninth entries of 4, 8, 12, … using 4n.
- A student says 3s − 2 = 15 when s = 7. Write the substitution and correction.
- Correct 2r + 1 = 29 for r = 8.
- Check 2j = 10 when j = 5. Is there an error?
- Evaluate 2f − 2g when f = 3, g = 1, and 2t + b when t = 4, b = 3.
- Evaluate h − (3 − n) when h = 5 and n = 6.
- Explain why 68 − (18 + 13) equals 68 − 18 − 13.
- Evaluate 5m + 3 for m = −2.
40; 28 + 54 = 82; 20 + 8 × 10 = 100.
Key Takeaways
• Arithmetic rules apply when letters are replaced by numbers. • Evaluate bracketed operations and products before the final sum. • Keep each term’s sign when swapping or grouping. • A number beside a letter indicates multiplication. • Replace negative values using brackets to preserve their signs. • A correct answer comes from a correct sequence of operations.