Expressions using Letter-Numbers · Lesson 4 of 11
Simplification of Algebraic Expressions
“See why like terms can be collected and why different-looking expressions can have the same value.”
• Derive the rectangle-perimeter expression by collecting equal side lengths. • Identify terms, coefficients, and like and unlike terms. • Explain simplification using repeated addition and the distributive property. • Use split rectangles to justify addition and subtraction of like terms. • Check that simplification preserves values after substitution.
Simpler Writing for the Same Quantity
Evaluating finds a number after values are supplied. Simplifying changes the expression’s form while leaving its value unchanged for every allowed assignment of its letters. We can often do this before the values are known. A rectangle’s perimeter provides a concrete example.
Let l be the rectangle’s length and b its breadth. Walking around its boundary gives l + b + l + b. Because this is a sum, regroup the two lengths and the two breadths: l + l + b + b. Two copies of l give 2l, and two copies of b give 2b. The perimeter is therefore 2l + 2b.
Problem
For l = 3 cm and b = 4 cm, compare l + b + l + b with 2l + 2b.
- 1.The original expression gives 3 + 4 + 3 + 4 = 14 cm.
- 2.The simplified expression gives 2 × 3 + 2 × 4 = 6 + 8 = 14 cm.
- 3.The agreement checks this case. The regrouping argument explains why the expressions agree for any rectangle side lengths.
Rewriting an expression in a simpler form using arithmetic rules, without changing its value for the allowed values of its letters.
Terms and Like Terms
To simplify, first see an expression as a sum of signed parts. In 5c + 3c − 2d + 7, those parts are 5c, 3c, −2d, and 7. A term can contain multiplication, even though the terms are joined by addition. The term 7 has no letter and is a number-only, or constant, term.
The numerical multiplier of a letter part. In 5c the coefficient is 5; in −2d it is −2. A single c means 1c, and −c means −1c.
Terms with the same letter part. In this chapter, 5c, 3c, and −c are like terms because each is a multiple of c. Number-only terms can also be combined with one another.
Terms with different letter parts, such as 18c and 11d. They cannot be collected into a single multiple of one letter without further information about those letters.
A coefficient tells us how many copies, or how much of a multiple, of a quantity appears. Five copies of c plus three copies of c make eight copies of c. Five copies of c plus three copies of d do not generally make eight copies of either one. The letters identify the kinds of quantities being combined.
Repeated Addition and the Distributive Property
A shop sells 5, 3, and 10 pencils on three successive days. If each pencil costs c rupees, the income is 5c + 3c + 10c. All three terms count multiples of the same price, so the coefficients can be added while the price c stays attached.
Problem
Pencils sold over three days: 5, 3, 10. Erasers sold: 4, 6, 1. Prices are c and d rupees respectively. Find the total income.
- 1.Pencil income is 5c + 3c + 10c = (5 + 3 + 10)c = 18c.
- 2.Eraser income is 4d + 6d + d = (4 + 6 + 1)d = 11d.
- 3.Total income is 18c + 11d rupees. The c terms and d terms stay separate because the two prices need not be equal.
- 4.For c = ₹50, pencil income is 18 × 50 = ₹900. If d = ₹10 as well, total income is 900 + 110 = ₹1010.
The distributive property works in both directions. It expands a multiple of a sum into separate products, or collects separate multiples into one product. An area picture helps make this connection visible: a rectangle can be measured as one whole region or as two adjacent regions with the same height.
Problem
A rectangle of height v units is split into widths 4 and 3 units. Express the total area in two ways.
- 1.The smaller rectangles have areas 4v and 3v square units.
- 2.Their total area is 4v + 3v.
- 3.The whole width is 4 + 3 = 7, so its area is 7v. Both count the same region: 4v + 3v = 7v.
Subtracting Like Terms
The same reasoning applies when part of a quantity is removed. A rectangle of width 12 and height n has a strip of width 4 and the same height removed. The area that remains can be found by subtracting areas or by calculating the remaining width first.
Problem
Find the remaining area after removing a width-4 strip from a width-12 rectangle, both of height n.
- 1.Whole area is 12n square units; removed area is 4n square units.
- 2.Remaining area is 12n − 4n = (12 − 4)n.
- 3.Because the remaining width is 8, the result is 8n square units. Subtracting coefficients matches the geometry.
Problem
Simplify 7p − 3p + 2q − q + 6 − 9.
- 1.The p terms give (7 − 3)p = 4p.
- 2.The q terms give (2 − 1)q = q. Number-only terms give 6 − 9 = −3.
- 3.Combine the separate groups: 4p + q − 3. No values of p or q were needed.
Simplification does not require all terms to disappear into one. In an expression with independent p and q terms, a combined form may still contain both letters. Conversely, cancellation can remove a letter completely: 3b − 2b − b = (3 − 2 − 1)b = 0. Thus the number of letters you first see does not always equal the number of terms left at the end.
Do not simplify 18c + 11d to 29cd or 29c. Adding unlike terms is not the same as multiplying their letters. Also, the sign belongs to the coefficient: 12n − 4n uses 12 + (−4), not 12 + 4.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
Which is a like term of 5c?
What is the simplified rectangle perimeter l + b + l + b?
Simplify 12n − 4n.
What is the coefficient of the term −d?
What happens to 18c + 11d when c and d have no stated relationship?
3c has the same letter part c.
Practice Problems
- Simplify p + p + p + p and p + p + p + q.
- Simplify p + q + p − q and p − q + p − q.
- Simplify p + q − p + q and p − q − p − q.
- Simplify 2d − d − d − d and 2d − d − d − c.
- Simplify 2d − d − c − c.
- Use the three-day sales data to compare 5c + 3c + 10c with 18c for c = 2 and c = 7.
- Find the area of the two-part rectangle when v = 6.
- Find the retained area in the subtraction diagram when n = 5.
- Explain why 3b − 2b − b is zero for every value of b.
4p and 3p + q. The q term is unlike the p terms.
Key Takeaways
• Simplifying preserves an expression’s value without needing assigned letter values. • Terms must be read together with their signs. • A coefficient is the numerical multiplier of a letter part. • Collect like terms by adding or subtracting their coefficients. • Keep unlike terms separate unless extra information justifies combining them. • The distributive property connects repeated addition, area, and collection of terms.