Practice Problems

Hints

  1. The simplified expression is ‍\(7z\).

To combine like terms, we find all of the terms with the same variable to the same power. Then we add their coefficients.

That gives us an equivalent expression with fewer terms.

Hint: We can commute, associate, and distribute the terms once we rewrite all subtraction as addition of the opposite.

We rewrite as addition. Then we combine the z terms:

\( 4z-(-3z)+ = \underbrace{\operatorname{4z}}_{\text{term}}+\underbrace{\operatorname{3z}}_{\text{term}} \)

\(=(4+3)z\)

\(=7z\)

  1. The simplified expression is ‍\(-3k+10\)

    Combine the numeric terms:

    \( 3k+8+2 = -3k+10\)

  2. The simplified expression is ‍\(2n+2\)

    Combine the ‍n terms:

    \(-n+(-3) + 3n+5 = (-1+3)n-3+5 = 2n -3+5\)

    Combine the numeric terms:

    \(2n-3+5 = 2n+2\)

  3. The simplified expression is ‍\(-6r+5\).

    We rewrite as addition. Then we combine the r terms:

    \( -4r -2r+5 = \underbrace{\operatorname{-4r}}_{\text{term}}+\underbrace{\operatorname{-2r}}_{\text{term}}+\underbrace{\operatorname{5}}_{\text{term}} \)

    \(=(-4-2)r+5\)

    \(=-6r+5\)


  4. The simplified expression is ‍\(7k\).

    We rewrite as addition. Then we combine the k terms:

    \(k-(-8k) = \underbrace{\operatorname{-k}}_{\text{term}}+\underbrace{\operatorname{8k}}_{\text{term}} \)

    The coefficient of \(-k\) is \(-1 \cdot k = -k\)

    \(=(-1+8)k\)

    \(=7k\)

  5. The simplified expression is ‍\(-6p-10\).

    We rewrite as addition. Then we combine the p terms:

    \(-12-6p+2 =-12 +(-6p)+2\)

    \(=-6p+2+(-12)\)

    \(=-6p-10\)

  6. The simplified expression is ‍\(-2r+8\)

    Combine the r terms:

    \(2r+1+(-4r)+7 = (2-4)r + 1+7\)

    \(=-2r+1+7\)

    Combine the numeric terms:

    \(-2r+1+7 = -2r+8\)

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