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C2.1 · Combine first-degree monomials and binomials with integer coefficients

Learn to combine first-degree monomials and binomials with integer coefficients through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Algebra

Ontario Grade 8 Mathematics — C2.1

Imagine that each blue tile stands for the same unknown number, called xx. Three blue tiles and two more blue tiles make five blue tiles. In symbols, this is 3x+2x=5x3x+2x=5x. Small plain tiles can stand for units: four plain tiles and seven negative units have a total of negative three units. The key idea is to combine only matching kinds of tiles. This lesson uses that idea to combine monomials and binomials with integer coefficients.

What you will learn

  • Identify terms, monomials, binomials, and like terms.
  • Combine first-degree monomials and binomials with integer coefficients.
  • Keep track of signs when adding or subtracting a binomial.

1. Recognize the parts

A variable is a letter that stands for a number. We will use letters such as xx and yy. A coefficient is the number multiplying a variable. In 5x5x, the coefficient is 55. The coefficient of −3x-3x is −3-3, because the sign belongs with the number.
A constant is a number with no variable, such as −4-4. A term is one part of an expression, separated by plus or minus signs. For example, the terms in 6x−26x-2 are 6x6x and −2-2. Keep the minus sign with the term that follows it.
A monomial is an expression with one term, such as 7x7x or −5-5. A binomial is an expression with two terms, such as 3x+43x+4 or x−yx-y. In this lesson, first-degree variable terms use a variable to the first power, such as xx or yy. We do not need to write the power of one.
Like terms have the same variable part. For example, 3x3x and −8x-8x are like terms. The constants 44 and −9-9 are also like terms. But 3x3x and 3y3y are not like terms, and neither are 3x3x and 33. The variable parts differ, or one term has no variable.
  • A term includes its sign.
  • Like terms have the same variable part, or are both constants.
  • A monomial has one term; a binomial has two.

2. Combine matching terms

Think of xx-terms as identical tiles. Three xx-tiles and two more xx-tiles make five xx-tiles. The variable part stays xx, while the coefficients are added. This gives 3x+2x=5x3x+2x=5x.
The same idea works with negative coefficients. If five xx-tiles are followed by taking away eight xx-tiles, the total is negative three xx-tiles. So 5x−8x=−3x5x-8x=-3x. For constants, use the usual integer addition and subtraction. For example, 4−7=−34-7=-3.
When adding expressions, keep each term's sign and gather like terms. When subtracting a binomial, change the sign of every term in that binomial. The minus sign applies to the whole binomial, not just its first term. For example, subtracting 2x−32x-3 changes its terms to −2x+3-2x+3.
After combining, unlike terms stay separate. For instance, 2x+52x+5 cannot be combined into one term because 2x2x has a variable and 55 does not. A clear result shows the different kinds of terms separately.
ax+bx=(a+b)xax+bx=(a+b)x
  • Add or subtract coefficients of like terms.
  • Keep the shared variable part unchanged.
  • When subtracting a binomial, change both term signs.

3. Sort, combine, and check

A reliable method has three moves. First, write the terms clearly and keep their signs. Next, sort terms by kind: for example, xx-terms together and constants together. Finally, combine within each group and check that unlike terms remain separate.
Sorting is like putting matching tiles in the same pile. It helps you see which coefficients belong together. It does not change the value of the expression; it only makes the groups easier to read.
For subtraction, handle the signs before sorting. Rewrite the subtracted binomial with both signs changed. Then gather like terms. This order makes it less likely that a negative sign will be missed.
A final check is to count the kinds of terms in the answer. If the original has variable terms and constants, the result may still have both. Do not combine them just to make the expression look shorter.
  • Keep signs attached to their terms.
  • Sort by matching variable part before combining.
  • Check that unlike terms have not been combined.

4. Use the method in practice

In each example, notice which terms match before doing any arithmetic. The coefficient can be positive, negative, or zero after combining. If the coefficients cancel, the variable terms make zero and can be left out of the final expression.
Read each final expression by its term types. A result with one variable term and one constant is still written as two separate terms. A result with only a variable term has no constant to include.
After the examples, try the quick check. For each choice, examine both the coefficient and the variable part. A correct answer must combine matching terms and preserve the effect of every sign.
  • The coefficient includes its sign.
  • A zero total for a group means that group does not appear in the answer.
  • Keep different variable parts and constants separate.

Worked example

Add two binomials

Simplify (3x+4)+(2x−7)(3x+4)+(2x-7).
  1. Sort matching terms
    The expression has two xx-terms and two constants. Keep each term's sign as you place matching terms together.
    3x+2x+4−73x+2x+4-7
  2. Combine each group
    Add the coefficients of the xx-terms. Then combine the constants using integer subtraction. Keep the variable part xx with its combined coefficient.
    (3+2)x+(4−7)(3+2)x+(4-7)
Answer: 5x−35x-3
Check: The two variable terms make five xx-tiles, and the constants total negative three. The answer has one variable term and one constant, as expected.

Worked example

Subtract a binomial

Simplify (6y−2)−(3y+5)(6y-2)-(3y+5).
  1. Change both signs
    The subtraction sign applies to the entire second binomial. Change the sign of 3y3y and the sign of 55 before combining.
    6y−2−3y−56y-2-3y-5
  2. Combine matching terms
    Combine the yy-terms together and the constants together. The coefficients 66 and −3-3 total 33. The constants −2-2 and −5-5 total −7-7.
    (6−3)y+(−2−5)(6-3)y+(-2-5)
Answer: 3y−73y-7
Check: Both terms in the subtracted binomial changed sign. Combining only matching terms gives a yy-term and a constant.

Worked example

Combine a monomial and a binomial

Simplify (−4a)+(7−6a)(-4a)+(7-6a).
  1. Group the like terms
    The monomial −4a-4a matches the term −6a-6a. The constant 77 has no variable, so it stays in a separate group.
    −4a−6a+7-4a-6a+7
  2. Combine the coefficients
    Add the coefficients of the matching aa-terms. Since both coefficients are negative, their total is negative ten. Keep the constant unchanged.
    (−4−6)a+7(-4-6)a+7
Answer: −10a+7-10a+7
Check: The two aa-terms combine to −10a-10a. The constant remains 77 because it is not like an aa-term.

Common mistakes and how to avoid them

Combining 3x3x and 44 to get 7x7x.
Correction: 3x3x has a variable part, but 44 does not. They are unlike terms, so keep them separate.
Changing the sign of only the first term when subtracting a binomial.
Correction: The subtraction sign applies to every term in the binomial. Change both signs before combining.
Treating the coefficient of −5x-5x as 55.
Correction: The coefficient includes its sign. The coefficient of −5x-5x is −5-5.

Lesson summary

  • A monomial has one term; a binomial has two terms.
  • Like terms have the same variable part, or are both constants.
  • Combine the integer coefficients of like terms and keep the variable part.
  • When subtracting a binomial, change the sign of each term in it.
  • Do not combine unlike terms.

Check your understanding

Question 1

Simplify 7x−3x+27x-3x+2.
  1. 4x+24x+2
  2. 4x4x
  3. 10x+210x+2
  4. 4x+2x4x+2x
Show answer and explanation
4x+24x+2
Combine the like terms 7x7x and −3x-3x to get 4x4x. The constant 22 stays separate.

Question 2

Simplify (2y+6)−(5y−1)(2y+6)-(5y-1).
  1. −3y+5-3y+5
  2. −3y+7-3y+7
  3. 7y+57y+5
  4. 3y+73y+7
Show answer and explanation
−3y+7-3y+7
Change both signs in the subtracted binomial to get 2y+6−5y+12y+6-5y+1. The yy-terms total −3y-3y, and the constants total 77.

Question 3

Simplify (4a)+(3a+2)(4a)+(3a+2).
  1. 7a+27a+2
  2. 7a7a
  3. a+2a+2
  4. 12a+212a+2
Show answer and explanation
7a+27a+2
The coefficients of the matching aa-terms add to 77. The constant 22 remains separate.

Key terms

Variable
A letter that stands for a number.
Coefficient
The number multiplying a variable, including its sign.
Constant
A number with no variable.
Term
One part of an expression, separated by plus or minus signs.
Monomial
An expression with one term.
Binomial
An expression with two terms.
Like terms
Terms with the same variable part, or constants that can be combined with other constants.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation C2.1. It is a study resource, not an official curriculum publication.

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