Expressions using Letter-Numbers | CBSE Class 7 Maths Notes

This note covers letter-numbers, algebraic expressions, formulas, substitution, multiplication notation, like and unlike terms, simplifying expressions, removing brackets, comparing expressions, and using algebra to explain repeating designs, calendar relationships and matchstick patterns.

What are letter-numbers and algebraic expressions?

Definition: A letter-number is a letter used to represent a number. An algebraic expression is a mathematical expression containing letter-numbers.

A mathematical relationship describes how quantities are connected. Shabnam is 3 years older than Aftab, so her age is found by adding 3 to his age. The relationship remains useful even when their ages change.

Let a represent Aftab’s age in years and s represent Shabnam’s age in years. Then a + 3 represents Shabnam’s age, and s = a + 3. Here, + means addition and = means that the quantities on its two sides are equal.

Letters are shorthand for quantities, with their meanings stated in the problem. Usually, letters or short phrases are used for this purpose. The choice of letter is convenient; its meaning comes from the quantity it represents.

How can the same relationship work in reverse?

Aftab is 3 years younger than Shabnam, so a = s − 3. The symbol − means subtraction here. Adding 3 finds the older person’s age; subtracting 3 finds the younger person’s age.

Worked example 1. Shabnam is 3 years older than Aftab. Find her age when Aftab is 23 years old.

Answer: Replace a by 23 in s = a + 3. Thus s = 23 + 3 = 26. Shabnam is 26 years old.

Substitution means replacing a letter-number with a specified number. The expression describes the relationship before substitution; afterwards, arithmetic gives a particular value. Keep the meaning and unit of the quantity attached to the result.

How do situations become algebraic expressions?

Begin by describing the relationship in words. Decide which quantities are given, which quantity is needed, and which operations connect them. An operation is an action such as adding, subtracting or multiplying numbers.

How can a shopping cost be represented?

Ketaki buys coconuts at ₹35 each and jaggery at ₹60 per kilogram. The symbol ₹ denotes rupees, while kg abbreviates kilogram. Let c be the number of coconuts and j the quantity of jaggery in kilograms. The multiplication symbol × means “times”.

Add the two costs to obtain the total amount in rupees: c × 35 + j × 60. Each quantity is multiplied by its own price before the costs are added.

Worked example 2. Find Ketaki’s cost for 10 coconuts and 5 kg of jaggery, at ₹35 per coconut and ₹60 per kilogram of jaggery.

Answer: The coconut cost is 10 × ₹35 = ₹350. The jaggery cost is 5 × ₹60 = ₹300. Therefore, the total is ₹350 + ₹300 = ₹650.

How do fixed and changing quantities differ?

For Munirathna’s pipes, let k represent the length in metres of the pipe added to an existing 20-metre pipe. The combined length is 20 + k metres. Here 20 stays fixed while k represents the other length.

For Venkatalakshmi’s flour mill, let y be the grain quantity in kilograms. Starting the initially switched-off machine takes 10 seconds, and grinding takes 8 seconds per kilogram. The total time is 10 + 8 × y seconds: the starting time is included once.

How do formulas describe perimeters and repeated shapes?

Mathematical relationships written concisely using letter-numbers are often called formulas. A formula can be applied to different given values. It records the relationship without needing a separate verbal explanation for every calculation.

The perimeter of a shape is the total length of its boundary. A square has four equal sides. Let q be its side length, measured in centimetres, abbreviated cm. Its perimeter is 4 × q centimetres.

Worked example 3. Find the perimeter of a square with a side length of 7 cm.

Answer: Use four times the side length. Substituting q = 7 gives 4 × 7 = 28. The perimeter is 28 cm.

A regular polygon is a polygon whose side lengths and angle measures are all equal. A polygon is a closed shape made from straight sides. Equal sides allow a perimeter to be expressed as a repeated addition or a multiplication.

What relationship describes the L-shaped matchsticks?

Parthiv’s L-shaped pattern uses two matchsticks for each L. Let n represent the number of Ls. Then the number of matchsticks is 2 × n. The letter represents a count here, rather than a length or age.

The drawing shows groups containing one, two and three Ls. Each L consists of one upright matchstick and one horizontal matchstick.

See Fig. 4.2 in your NCERT textbook

For 5 Ls, the expression becomes 5 × 2; for 7 Ls, it becomes 7 × 2; and for 45 Ls, it becomes 45 × 2. The same rule describes every group in this pattern. Count what one repeated unit needs before multiplying by the number of units.

How are expressions evaluated using arithmetic rules?

To evaluate an expression is to find its numerical value. A sum is the result of addition. A term is one of the parts added when an expression is written as a sum. A term can itself contain multiplication, so it may need calculation before the terms are added.

Brackets, such as ( ), group an expression so it can be treated together. A minus sign attached to a number indicates a negative number, a number below zero.

Worked example 4. Evaluate 23 − 10 × 2.

Answer: Write the expression as 23 + (−10 × 2). The second term is −20. Hence the value is 23 + (−20) = 3.

In the calculation, the multiplication belongs to the second term. Subtracting 10 from 23 before multiplying would change the expression.

Property: Swapping and grouping terms preserves the value

Swapping means changing the order of terms being added. Grouping means collecting terms conveniently for addition. Keep each term’s sign when moving it. These operations do not change the value of the sum.

For example, 83 + 28 − 13 + 32 can be grouped as (83 − 13) + (28 + 32). This gives 70 + 60 = 130. Pairing the terms makes the calculation easier while preserving their contributions.

Property: Subtracting a bracket subtracts every term inside

In 68 − (18 + 13), either calculate the bracket first, giving 68 − 31 = 37, or remove it as 68 − 18 − 13 = 37.

The arithmetic rules also apply when letters represent numbers. Once values are substituted, evaluate the resulting arithmetic expression using its terms and brackets. In particular, retain brackets around substituted negative values so that their signs remain clear.

What does omitting the multiplication sign mean?

In algebraic notation, a number written immediately before a letter indicates multiplication. If n represents a number, the expression 4n means 4 × n. The standard shortened form places the number first, followed by the letter. It does not join digits into a larger number.

How does the notation describe a sequence?

A sequence is an ordered list of numbers. In 4, 8, 12, 16, 20, 24, 28, and so on, the entries are multiples of 4 in increasing order. A multiple is a number obtained by multiplying the given number by an integer.

An integer is a whole number, its negative, or zero. For this sequence, let n represent a positive position number. Its nth term, meaning the entry in position n, is 4n. For example, the third term is 4 × 3 and the 29th term is 4 × 29.

Worked example 5. Evaluate 7k when the letter-number k has value 4.

Answer: The expression 7k means 7 × k. Replacing k by 4 gives 7 × 4 = 28.

Worked example 6. Evaluate 5m + 3 when the letter-number m has value 2.

Answer: Replace m by 2, keeping multiplication between 5 and the substituted value. Then 5 × 2 + 3 = 10 + 3 = 13.

Note: Multiplication remains present even when its symbol is omitted. If d represents a number and d = 6, then 3d is 3 × 6 = 18, not 36.

Read an expression aloud before evaluating it. “Five times m, plus three” keeps the multiplication separate from the addition and helps preserve the intended calculation.

How do like terms simplify an expression?

Simplification rewrites an expression in a simpler form without changing its value. Let l be a rectangle’s length, b its breadth, and p its perimeter, all in the same length unit. Adding its sides gives p = l + b + l + b.

Swap and group the repeated lengths: l + l + b + b. Since l + l = 2l and b + b = 2b, the simplified perimeter formula is p = 2l + 2b. With l = 3 and b = 4, both forms give 14.

Property: The distributive property connects a multiple of a sum with a sum of multiples

A multiplier is the number by which a quantity is multiplied. The distributive property allows a multiplier to act on every term inside a bracket. It also allows repeated multiples of the same letter-number to be collected. The letter-number keeps representing the same quantity throughout the calculation.

In a shop example, let c be the price per pencil and d the price per eraser, both in rupees. The quantities sold are:

Worked example 7. Simplify the earnings from the pencil and eraser sales in the table.

Answer: Pencil earnings are 5c + 3c + 10c = (5 + 3 + 10)c = 18c. Eraser earnings are 4d + 6d + d = 11d. The combined earnings are 18c + 11d rupees.

Like terms in these expressions involve the same letter-number, such as 5c, c and 10c. Unlike terms, such as 18c and 11d, involve different letter-numbers. The latter cannot be combined into a single term by adding their numerical multipliers.

How do areas make simplification visible?

Area measures the surface enclosed by a shape. A rectangle’s area is its length multiplied by its breadth. When lengths are measured in a common unit, area is measured in square units, the area units represented by unit squares.

Let v denote the common height of two adjoining rectangles, in length units. Their widths are 4 and 3 units. The complete rectangle can be measured directly or by adding the areas of its two parts.

A rectangle has a vertical division. Its common height is labelled v, and the bottom widths of the two parts are labelled 4 and 3.

Reference: NCERT Class 7, page 89

Worked example 8. Find the total area of adjoining rectangles of height v and widths 4 and 3.

Answer: Adding the parts gives 4v + 3v square units. Using the full width gives (4 + 3)v = 7v square units. Thus 4v + 3v = 7v.

How does subtracting an area give another simplification?

In a second rectangle, n denotes the height in length units. The full width is 12 units and the right-hand part has width 4 units. The remaining width is 12 − 4, so its area is 8n square units.

The outer rectangle has corners A, B, C and D. A dashed line EF separates the right-hand rectangle EBCF. The full bottom length is 12, the height is n, and the right-hand width is 4.

Reference: NCERT Class 7, page 90

The letters A to F name points in the drawing; they are not letter-numbers in this calculation. The remaining rectangle AEFD has area 12n − 4n = 8n. This agrees with using its own side lengths and explains the subtraction of like terms geometrically.

How are algebraic expressions added and subtracted?

Work with complete terms, including their signs. After removing brackets correctly, collect like terms. A bracket preceded by a minus sign requires particular care: subtract the whole expression inside, rather than just its first term.

How is the amount returned on furniture handled?

Let x be the number of chairs and y the number of tables rented. The amounts paid initially and returned later are:

Worked example 9. Find the total amount paid after the returns for x chairs and y tables using these amounts.

Answer: Initially, 40x + 75y rupees are paid; 6x + 10y rupees are returned. Subtracting gives (40x + 75y) − (6x + 10y) = 40x + 75y − 6x − 10y = 34x + 65y rupees.

Both returned amounts reduce the payment. The minus sign before the second bracket therefore affects both 6x and 10y. Group the chair terms together and the table terms together after removing the bracket.

How are scores from several rounds combined?

For Charu’s quiz, let p represent the score awarded for a correct answer and q the penalty deducted for an incorrect answer. Her three round scores are 7p − 3q, 8p − 4q, and 6p − 2q.

Add the round scores, grouping the p terms and the negative q terms. The total becomes (7 + 8 + 6)p − (3 + 4 + 2)q = 21p − 9q. Both letters retain their quiz meanings throughout.

If p = 4 and q = 1, the first round gives 7 × 4 − 3 × 1 = 25. A penalty is the amount subtracted for an incorrect answer, so its sign must remain negative when the expressions are added.

How can we decide whether two expressions are equal?

Equal expressions take the same value when their letter-numbers are replaced by the same numbers. Different appearances do not necessarily mean different values. Applying valid arithmetic rules can show that one form simplifies to another.

Worked example 10. Simplify 4(x + y) − y, where x and y represent numbers.

Answer: Distribute 4 across both terms: 4(x + y) − y = 4x + 4y − y. Combining the y terms gives 4x + 3y.

The original expression and 4x + 3y are equal because the simplification uses rules that apply to the numbers represented. The bracket cannot simply disappear while leaving its multiplier attached to one term.

Why are multiplication and addition different?

Let u represent a number. The expression 5u means five times u, whereas 5 + u means five more than u. These operations give different results for most values of u. The two operations have different meanings.

For the given value u = 2, the expressions give 10 and 7 respectively. This disagreement shows that they are not equal expressions. A single matching value would not establish equality for every value of the letter-number.

How does a bracket change the meaning?

Let y represent a number. Then 10y − 3 means subtract 3 after multiplying y by 10. In contrast, 10(y − 3) means multiply the whole difference y − 3 by 10.

When y = 2, the first expression gives 17 and the second gives −10. Read the bracketed group as a unit before calculating. Operation order, meaning the order in which the calculation is performed, is part of the expression’s meaning.

How do letter-numbers describe number machines and repeating designs?

A number machine represents a rule that performs the same operations on its inputs. An input is a number supplied to the rule; the output is the result. State the rule in words before turning it into an expression.

For the rule “twice the first number minus the second”, let a be the first input and b the second. The expression is 2a − b. With first input 5 and second input 2, it gives 2 × 5 − 2 = 8.

How are positions in a repeating design described?

Somjit’s saree-border pattern repeats three designs. Call these Design A, Design B and Design C; these letters name designs. Let n represent the occurrence number of a particular design, starting with its first appearance.

Design C appears at positions 3, 6, and further multiples of 3. Its nth appearance is at 3n. Design B appears one place before the corresponding C, at 3n − 1. Design A appears two places before it, at 3n − 2.

The quotient is the whole-number result of division here, and the remainder is the amount left over. Dividing a position number by 3 helps identify which design occupies that position.

A remainder of 0 corresponds to C; a remainder of 2 corresponds to B; and a remainder of 1 corresponds to A. Thus positions 99, 122 and 148 contain C, B and A respectively. Distinguish the occurrence number from the position number in the complete pattern.

How does algebra explain calendar patterns?

In a calendar grid, moving one cell right adds 1, while moving one row down in the same column adds 7. A 2 × 2 block means two adjacent rows and two adjacent columns containing four dates.

A diagonal joins opposite corners of this block. In the block with top row 12, 13 and bottom row 19, 20, the diagonal pairs are 12 with 20, and 13 with 19. Their sums are equal.

How can the equality be shown generally?

Imagine continuing the calendar numbers into endless rows. Checking examples cannot examine every possible block. Instead, let a be the top-left number of any complete 2 × 2 block in this extended grid.

A four-cell grid has a and a + 1 in its top row, with a + 7 and a + 8 directly beneath them.

Reference: NCERT Class 7, page 99

Worked example 11. Compare the two diagonal sums in this general calendar block.

Answer: One diagonal gives a + (a + 8) = 2a + 8. The other gives (a + 1) + (a + 7) = 2a + 8. Therefore, both sums are equal for any value of a in the pattern.

What happens in a cross-shaped group?

A cross-shaped group contains a centre and its immediate neighbours above, below, left and right. The illustrated group has centre 15, with 8 above, 22 below, 14 to the left and 16 to the right.

Let a now denote the centre number. The neighbours are a − 7, a + 7, a − 1 and a + 1. Their differences from the centre cancel when added, so the total is 5a, five times the centre. This explains why the pattern continues.

How can a matchstick pattern predict a later step?

In the connected-triangle pattern, Step 1 contains one triangle and 3 matchsticks. Each following step adds a triangle using two additional matchsticks. The step number identifies the stage of the growing pattern.

Four stages show one, two, three and four joined triangles. Neighbouring triangles share a side, with horizontal sticks along the top or bottom and sloping sticks between them.

Reference: NCERT Class 7, page 100

Why does the rule contain one fewer addition?

Let y represent the step number. Step 1 already supplies the first 3 sticks, so reaching Step y requires y − 1 additions of two sticks. This gives 3 + 2 × (y − 1).

Worked example 12. Simplify the matchstick rule 3 + 2 × (y − 1).

Answer: Multiply both terms inside the bracket by 2, obtaining 3 + 2y − 2. Combine the numerical terms 3 and −2. The simplified expression is 2y + 1.

The two expressions describe the same count. The first records the starting triangle followed by repeated additions; the second gives a shorter calculation for any specified step. Neither requires drawing every earlier stage.

In this drawing, the sticks can also be separated by orientation, meaning the direction in which they lie. At Step 2, two sticks are horizontal and three are sloping. Counting different parts of one arrangement can produce different-looking expressions for the same total.

When forming a general rule, identify what changes and what stays fixed. Then explain why the expression matches the construction. A numerical pattern becomes more useful when its formula also explains how it arises.

Glossary

• Letter-number — A letter representing a number, with its meaning specified by the mathematical situation.
• Algebraic expression — A mathematical expression containing letter-numbers that represent numerical quantities in a relationship.
• Formula — A concise expression of a mathematical relationship using numbers, operations and letter-numbers.
• Substitution — Replacing a letter-number by a specified numerical value before carrying out the calculation.
• Term — One part of an expression written as a sum, kept with its sign.
• Like terms — Terms involving the same letter-number in these expressions, which can be combined by addition.
• Unlike terms — Terms involving different letter-numbers here, which cannot be combined by adding their numerical multipliers.
• Simplification — Rewriting an expression in a simpler form while preserving the value it represents.
• Distributive property — The rule that a multiple of a sum equals the sum of the corresponding multiples.
• Equal expressions — Expressions giving the same values when their letter-numbers are replaced by the same numbers.
• Perimeter — The total boundary length of a shape, found by adding its side lengths.
• Regular polygon — A closed straight-sided shape with all side lengths equal and all angle measures equal.
• Sequence — An ordered list of numbers whose entries are identified by their positions.
• Remainder — The amount left after division when the quotient is taken as a whole number.

Common errors and misconceptions

• Misconception: If d = 6, then 3d means 36. Correct: The adjacent number and letter indicate multiplication: 3d = 3 × 6 = 18.
• Misconception: If a = −4, then 10 − a is 6. Correct: Substitute the negative value with its sign: 10 − (−4) = 14.
• Misconception: For numbers a and b, 3a + 2b simplifies to 5. Correct: The terms contain different letter-numbers and cannot be combined in this way.
• Misconception: For a number p, 6(p + 2) becomes 6p + 8. Correct: Multiply both bracketed terms by 6: the result is 6p + 12.
• Misconception: For numbers x and y, (4x + 3y) − (3x + 4y) becomes x + y. Correct: Subtract both terms in the second bracket to obtain x − y.
• Misconception: The expressions 5u and 5 + u mean the same thing, where u is a number. Correct: They represent multiplication and addition, giving different results for most values of u.
• Misconception: For a number z, 5 − (2 − 6z) simplifies to 3 − 6z. Correct: Subtracting the negative term gives 5 − 2 + 6z = 3 + 6z.
• Misconception: Matching diagonal sums in one calendar block establishes the pattern for every block. Correct: Use a general top-left value a; both diagonal sums simplify to 2a + 8.

Exam-style questions with model answers

Q1. Shabnam is 3 years older than Aftab. Let a be Aftab’s age and s be Shabnam’s age, both in years. Write their age relationship and find Shabnam’s age when Aftab is 23. [2 marks]

1. Adding 3 to Aftab’s age gives Shabnam’s age, so the relationship is s = a + 3.
2. Substituting a = 23 gives s = 23 + 3 = 26. Shabnam is 26 years old.

Q2. Coconuts cost ₹35 each and jaggery costs ₹60 per kilogram. Ketaki buys 10 coconuts and 5 kilograms of jaggery. Find the two costs and the total payment. [3 marks]

1. The coconut cost is the number of coconuts multiplied by the price of each coconut: 10 × ₹35 = ₹350.
2. The jaggery cost is the quantity in kilograms multiplied by the price per kilogram: 5 × ₹60 = ₹300.
3. Add the two costs because the purchase includes both items. The total payment is ₹350 + ₹300 = ₹650.

Q3. The letter-numbers x and y represent numbers. Simplify 4(x + y) − y, explaining how the bracket and like terms are handled. [3 marks]

1. The multiplier 4 applies to both terms inside the bracket. By the distributive property, 4(x + y) becomes 4x + 4y.
2. Retain the subtraction outside the original bracket, giving 4x + 4y − y. The terms 4y and −y are like terms.
3. Combine them as (4 − 1)y = 3y. Therefore, the simplified expression is 4x + 3y.

Q4. A shop charges ₹40 per chair and ₹75 per table initially. It returns ₹6 per chair and ₹10 per table when the furniture is returned. Let x and y be the numbers of chairs and tables rented. Derive and simplify the total amount paid after the returns. [5 marks]

1. The initial chair payment is 40x rupees and the initial table payment is 75y rupees. Together, the initial payment is 40x + 75y rupees.
2. The returned amounts are 6x rupees for the chairs and 10y rupees for the tables, giving a total return of 6x + 10y rupees.
3. Subtract the whole returned amount from the initial payment. This gives the expression (40x + 75y) − (6x + 10y).
4. Remove the second bracket by subtracting both terms. The expression becomes 40x + 75y − 6x − 10y.
5. Collect the chair terms and table terms separately: (40 − 6)x + (75 − 10)y = 34x + 65y rupees.

Q5. Charu’s scores in three quiz rounds are 7p − 3q, 8p − 4q and 6p − 2q. Here p is the score for a correct answer and q is the penalty deducted for an incorrect answer. Find her first-round score for p = 4 and q = 1, then simplify her total score in terms of p and q. [4 marks]

1. Substitute the given values into the first-round expression: 7p − 3q = 7 × 4 − 3 × 1.
2. Calculate the two contributions, 28 and 3. The first-round score is therefore 28 − 3 = 25.
3. The combined score is (7p − 3q) + (8p − 4q) + (6p − 2q). Group the p terms and the subtracted q terms separately.
4. The total simplifies to (7 + 8 + 6)p − (3 + 4 + 2)q = 21p − 9q.

Q6. In an extended calendar grid, adjacent numbers increase by 1 to the right and by 7 down a column. A complete 2 × 2 block has top-left number a. Describe its other entries and prove that the diagonal sums are equal. [5 marks]

1. The top-right entry is a + 1, because it is one cell to the right of the given top-left entry a.
2. The bottom-left entry is a + 7. The bottom-right entry is a + 8, because it is one cell below and one cell right of a.
3. The diagonal through the top-left entry has sum a + (a + 8). Opening the bracket and combining the a terms gives 2a + 8.
4. The other diagonal has sum (a + 1) + (a + 7). Combining the two a terms and the numbers 1 and 7 also gives 2a + 8.
5. Both diagonal sums simplify to the same expression. Therefore, their equality holds for any complete block following these calendar-grid rules.

Q7. A connected-triangle matchstick pattern starts with 3 sticks at Step 1. Each next step adds 2 sticks. Let y be the step number. Derive two equal expressions for the number of sticks and find the count at Step 5. [4 marks]

1. The first step already contains 3 sticks. From Step 1 to Step y, there are y − 1 additions of two sticks.
2. The expression recording the starting amount and additions is therefore 3 + 2 × (y − 1).
3. Apply the distributive property: 3 + 2y − 2 = 2y + 1. This is a second, equal expression.
4. At Step 5, substitute y = 5: 2 × 5 + 1 = 11. The pattern uses 11 matchsticks.

Q8. For a letter-number u, compare the meanings of 5u and 5 + u. Use u = 2 to decide whether they are equal expressions. [2 marks]

1. The expression 5u means five times u, while 5 + u means five more than u.
2. At u = 2, their values are 10 and 7. Since these differ, the expressions are not equal.

Key takeaways

• Letter-numbers stand for numerical quantities, so define what each letter represents before using it in an expression.
• Describe a relationship in words first, then select the additions, subtractions or multiplications that express it.
• A number immediately before a letter indicates multiplication, even though the multiplication symbol has been omitted.
• Substitution gives an expression a numerical value while preserving its operations, brackets and the signs of substituted numbers.
• Swapping, grouping and the distributive property simplify expressions without changing the values they represent.
• Combine like terms, and remember that subtracting a bracket affects every term inside that bracket.
• Different-looking expressions can be equal; simplify them or compare their meanings carefully before deciding.
• General expressions explain why calendar relationships hold and predict positions or counts in repeating and growing patterns.

Test yourself

What makes an expression algebraic?

It contains letter-numbers representing numbers, allowing a mathematical relationship to be written without specifying every value.

If k represents a number, what does 7k mean?

It means 7 × k. The missing multiplication sign does not change the operation being performed.

What is 5m + 3 when the letter-number m equals 2?

Substitute 2 for m: 5 × 2 + 3 = 13, multiplying before adding the final 3.

Why can 18c + 11d, where c and d are different letter-numbers, not be collected into one like term?

The two terms contain different letter-numbers. Adding their numerical multipliers would not preserve the quantities represented.

In the repeating order A, B, C, where is the nth occurrence of Design B?

It is at position 3n − 1, one position before the corresponding occurrence of Design C.

A 2 × 2 calendar block increases by 1 rightwards and 7 downwards. If a is its top-left entry, what is its bottom-right entry?

The bottom-right entry is a + 8, combining an increase of 7 with an increase of 1.

What is the sum of a calendar cross with centre a and neighbours a − 7, a + 7, a − 1 and a + 1?

The sum is 5a: the positive and negative differences cancel, leaving five copies of the centre number.

Why is the matchstick expression 3 + 2(y − 1) equal to 2y + 1?

Distributing 2 gives 3 + 2y − 2, and combining 3 with −2 leaves 2y + 1.