Simplify Algebraic Expressions Using like terms - Grade 6

This page presents Grade 6 examples and practice questions on adding and subtracting like terms in algebra. Questions with detailed step-by-step solutions and clear explanations are provided at the bottom to help students master simplifying expressions.

Review: Like Terms and Simplifying

In an algebraic expression, like terms are all terms with the same variable raised to the same powers.

Examples: Like Terms

1) \( 2x , \; 4x , \; x , \; \text{and } 10x \) are all like terms with coefficients \( 2, \; 4, \; 1, \; \text{and } 10 \) respectively.

Note that the coefficients of like terms may be different but the power of the variable, \( x \) in this case must be the same.

2) In the algebraic expression: \[ 2x + 7x - 6 + 4x^{2} - 6x , \] the terms \( 2x, \; 7x, \; \text{and } -6x \) are all like terms since they have the same variable \( x \) raised to the same power \( 1 \).

Like terms are important because they can be added and subtracted, which leads to the simplification of algebraic expressions.

We add and/or subtract like terms by adding their coefficients.


Examples: Simplifying Algebraic Expressions

3) Simplify, by adding and subtracting, the expression: \[ 3 x + 10 + 5 x - 6 x - 4 \]

Given \[ 3 x + 10 + 5 x - 6 x - 4\] Use parentheses to put like terms together \[ = (3 x + 5 x - 6 x) + (10 - 4) \] identify coefficients and put variable out of parentheses (factoring) \[ = (3 + 5 - 6) x + (10 - 4) \] Add and/or subtract coefficients and numbers to simplify \[ = 2 x + 6 \]
4) Use distributive property , then add and subtract to simplify the expression \( 2(x - 3) + 3(x + 1) \)

Use distributive property to expand \( 2(x - 3) \) ann \( 3(x + 1) \) : \[ 2(x - 3) = (2)(x) + (2)(-3) = 2 x - 6 \] \[ 3(x + 1) = (3)(x) + (3)(1) = 3 x + 3 \] We now write the whole expression using the above results: \[ 2(x - 3) + 3(x + 1) = 2 x - 6 + 3 x + 3 \] Group like terms : \[ = (2 x + 3 x) + ( - 6 + 3 ) = ( 2 + 3) x + ( - 6 + 3 ) = 5 x - 3 \] Note that all real numbers are like terms because they can be added and subtracted.


Solve the following questions (solutions at the bottom of the page)


  1. What are the like terms and their coefficients in each of the following expressions?
    1. \( 7 x - 5 x + 6 + 3x \)
    2. \( 6 x + 11 x + 9 + x - 7 \)
    3. \( 3 x + 9 y + 9 x - y \)
    4. \( \dfrac{1}{3} x + (2/3) x - 3 \)

  2. Simplify the following expressions.
    1. \( 2 x + x \)
    2. \( 12 x - 5 x + 11 - 4 x \)
    3. \( 9 x + 11 x + 19 - x - 7 \)
    4. \( 4 x + 8 y + 10 x - 3 y \)
    5. \( x + 5 + 3 y + 6 x - x - 2 y + 6 \)
    6. \( \dfrac{1}{2} x + \dfrac{3}{2} x + 2 \)

  3. Simplify the following expressions and identify the ones that have the same terms (equivalent expressions).
    1. \( 3 x + 3 + 4 x + 2 \)
    2. \( 4 x + 2 a + 6 x + 6a \)
    3. \( x + 2 + 3 x + 3 x + 3 \)
    4. \( x - x + 3 y \)
    5. \( 7 a + 10 x + a \)
    6. \( 4 x + 3 y - 4 x \)

  4. Use the distributive property then simplify the following expressions.
    1. \( 3 (x + 1) + 5 x - 4 \)
    2. \( 3 (x - 1) + 2 (x + 2) + 5 x \)
    3. \( 4 (x + 2) + 2 (y + 2) + x + 3 y \)
    4. \( 3 (\dfrac{x}{3} + 1) + 2 (\dfrac{x}{2} - 1) + 3 \)

Solutions to the Above Questions

  1. Solution


    Like terms have the same variable raised to the same power. Hence
    1. In the expression \( 7x - 5x + 6 + 3x \), the terms \( 7x \), \( -5x \), and \( 3x \) are like terms. The coefficients of the like terms \( 7x \), \( -5x \), and \( 3x \) are \( 7 \), \( -5 \), and \( 3 \) respectively.
    2. In the expression \( 6x + 11x + 9 + x - 7 \), the terms \( 6x \), \( 11x \), and \( x \) are like terms; \( 9 \) and \( -7 \) are numbers and therefore like terms. The coefficients of the like terms \( 6x \), \( 11x \), and \( x \) are \( 6 \), \( 11 \), and \( 1 \) respectively. The numbers are \( 9 \) and \( -7 \).
    3. In the expression \( 3x + 9y + 9x - y \), there are two groups of like terms: 1) \( 3x \) and \( 9x \) are like terms. 2) \( 9y \) and \( -y \) are like terms. The coefficients of the like terms \( 3x \) and \( 9x \) in group (1) are \( 3 \) and \( 9 \) respectively. The coefficients of the like terms \( 9y \) and \( -y \) in group (2) are \( 9 \) and \( -1 \) respectively.
    4. In the expression \( \tfrac{1}{3}x + \tfrac{2}{3}x - 3 \), the terms \( \tfrac{1}{3}x \) and \( \tfrac{2}{3}x \) are like terms, and their coefficients are \( \tfrac{1}{3} \) and \( \tfrac{2}{3} \) respectively.

  2. Solution

    To simplify algebraic expressions, we first group and then add and/or subtract the coefficients of the like terms (see examples above).
    1. Given: \[ 2x + x \] Identify coefficients and factor: \[ (2 + 1)x \] Simplify: \[ 3x \]
    2. Given: \[ 12x - 5x + 11 - 4x \] Group like terms: \[ (12x - 5x - 4x) + 11 \] Factor \(x\): \[ (12 - 5 - 4)x + 11 \] Simplify: \[ 3x + 11 \]
    3. Given: \[ 9x + 11x + 19 - x - 7 \] Group like terms: \[ (9x + 11x - x) + (19 - 7) \] Factor \(x\): \[ (9 + 11 - 1)x + (19 - 7) \] Simplify: \[ 19x + 12 \]
    4. Given expression has two variables: \[ 4x + 8y + 10x - 3y \] Group like terms: \[ (4x + 10x) + (8y - 3y) \] Factor \(x\) and \(y\): \[ (4 + 10)x + (8 - 3)y \] Simplify: \[ 14x + 5y \]
    5. Given expression has two variables: \[ x + 5 + 3y + 6x - x - 2y + 6 \] Group like terms: \[ (x + 6x - x) + (3y - 2y) + (5 + 6) \] Factor \(x\) and \(y\): \[ (1 + 6 - 1)x + (3 - 2)y + (5 + 6) \] Simplify: \[ 6x + y + 11 \]
    6. Given expression has fractions: \[ \tfrac{1}{2}x + \tfrac{3}{2}x + 2 \] Group like terms: \[ \left(\tfrac{1}{2}x + \tfrac{3}{2}x\right) + 2 \] Factor \(x\): \[ \left(\tfrac{1}{2} + \tfrac{3}{2}\right)x + 2 \] Simplify: \[ \tfrac{4}{2}x + 2 = x + 2 \]


  3. Solution


    We first simplify the given expressions (see exercise 2 above)
    1. \(3x + 3 + 4x + 2 = (3x + 4x) + (3 + 2) = (3 + 4)x + (3 + 2) = 7x + 5\)
    2. \(4x + 2a + 6x + 6a = (4x + 6x) + (2a + 6a) = (4 + 6)x + (2 + 6)a = 10x + 8a\)
    3. \(x + 2 + 3x + 3x + 3 = (x + 3x + 3x) + (2 + 3) = (1 + 3 + 3)x + (2 + 3) = 7x + 5\)
    4. \(x - x + 3y = (1 - 1)x + 3y = 0x + 3y = 0 + 3y = 3y\)
    5. \(7a + 10x + a = (7a + a) + 10x = (7 + 1)a + 10x = 8a + 10x\)
    6. \(4x + 3y - 4x = (4x - 4x) + 3y = (4 - 4)x + 3y = 0x + 3y = 0 + 3y = 3y\)

    Conclusion
    The expressions in parts a) and c) are equivalent;
    the expressions in parts b) and e) are equivalent
    and the expressions in parts d) and f) are equivalent.

  4. Solution

    We first use the distributive property to take out the parentheses and then simplify.

    1. Given: \[ 3 (x + 1) + 5x - 4 \] Use distributive property to multiply: \[ = (3)(x) + (3)(1) + 5x - 4 \] Multiply and simplify: \[ = 3x + 3 + 5x - 4 \] Use parentheses to put like terms together: \[ = (3x + 5x) + (3 - 4) \] Identify coefficients and factor out \( x \): \[ = (3 + 5)x + (3 - 4) \] Simplify: \[ = 8x - 1 \]
    2. Given: \[ 3 (x - 1) + 2 (x + 2) + 5x \] Use distributive property to multiply: \[ = (3)(x) + (3)(-1) + (2)(x) + (2)(2) + 5x \] Multiply and simplify: \[ = 3x - 3 + 2x + 4 + 5x \] Use parentheses to put like terms together: \[ = (3x + 2x + 5x) + (-3 + 4) \] Identify coefficients and factor out \( x \): \[ = (3 + 2 + 5)x + (-3 + 4) \] Simplify: \[ = 10x + 1 \]
    3. Given: \[ 4 (x + 2) + 2 (y + 2) + x + 3y \] Use distributive property to multiply: \[ = (4)(x) + (4)(2) + (2)(y) + (2)(2) + x + 3y \] Multiply and simplify: \[ = 4x + 8 + 2y + 4 + x + 3y \] Use parentheses to put like terms together: \[ = (4x + x) + (2y + 3y) + (8 + 4) \] Identify coefficients and factor out variables: \[ = (4 + 1)x + (2 + 3)y + (8 + 4) \] Simplify: \[ = 5x + 5y + 12 \]
    4. Given: \[ 3 \left(\dfrac{x}{3} + 1\right) + 2 \left(\dfrac{x}{2} - 1\right) + 3 \] Use distributive property to multiply: \[ = (3)\left(\dfrac{x}{3}\right) + (3)(1) + (2)\left(\dfrac{x}{2}\right) + (2)(-1) + 3 \] Multiply and simplify: \[ = \dfrac{3x}{3} + 3 + \dfrac{2x}{2} - 2 + 3 \] Use parentheses to put like terms together: \[ = \left(\dfrac{3x}{3} + 3 + \dfrac{2x}{2}\right) + (3 - 2 + 3) \] Identify coefficients and factor out \( x \): \[ = \left(\dfrac{3}{3} + \dfrac{2}{2}\right) x + (3 - 2 + 3) \] Simplify: \[ = (1 + 1) x + 4 = 2x + 4 \]

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