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Multi-Step Equations with Like Terms

Add and subtract like terms as a step to solve equations.

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Multi-Step Equations with Like Terms

Multi-Step Equations with Like Terms

When we look at a linear equation we see two kinds of terms: those that contain the unknown variable, and those that don’t. When we look at an equation that has an \begin{align*}x\end{align*} on both sides, we know that in order to solve it, we need to get all the \begin{align*}x-\end{align*}terms on one side of the equation. This is called combining like terms. The terms with an \begin{align*}x\end{align*} in them are like terms because they contain the same variable (or, as you will see in later chapters, the same combination of variables).

Like Terms Unlike Terms
\begin{align*}4x, 10x, -3.5x,\end{align*} and \begin{align*}\frac{x}{12}\end{align*} \begin{align*}3x\end{align*} and \begin{align*}3y\end{align*}
\begin{align*}3y, 0.000001y,\end{align*} and \begin{align*}y\end{align*} \begin{align*}4xy\end{align*} and \begin{align*}4x\end{align*}
\begin{align*}xy, 6xy,\end{align*} and \begin{align*}2.39xy \end{align*} \begin{align*}0.5x\end{align*} and \begin{align*}0.5 \end{align*}

Using the Distributive Property of Multiplication

To add or subtract like terms, we can use the Distributive Property of Multiplication.

\begin{align*}3x + 4x &= (3 + 4)x = 7x \\ 0.03xy - 0.01xy &= (0.03 - 0.01)xy = 0.02xy\\ -y + 16y + 5y &= (-1 + 16 + 5)y = 10y\\ 5z + 2z - 7z &= (5 + 2 - 7)z = 0z = 0\end{align*}

To solve an equation with two or more like terms, we need to combine the terms first.

Solving for Unknown Values

1. Solve \begin{align*}(x + 5) - (2x - 3)=6\end{align*}.

There are two like terms: the \begin{align*}x\end{align*} and the \begin{align*}-2x\end{align*} (don’t forget that the negative sign applies to everything in the parentheses). So we need to get those terms together. The associative and distributive properties let us rewrite the equation as \begin{align*} x + 5 - 2x + 3 = 6\end{align*}, and then the commutative property lets us switch around the terms to get \begin{align*}x - 2x + 5 + 3 = 6\end{align*}, or \begin{align*}(x - 2x) + (5 + 3) = 6\end{align*}.

\begin{align*}(x - 2x)\end{align*} is the same as \begin{align*}(1 - 2)x\end{align*}, or \begin{align*}-x\end{align*}, so our equation becomes \begin{align*}-x + 8 = 6\end{align*}

Subtracting 8 from both sides gives us \begin{align*}-x = -2 \end{align*}.

And finally, multiplying both sides by -1 gives us \begin{align*}x = 2 \end{align*}.

2. Solve \begin{align*}\frac{x}{2} - \frac{x}{3} = 6 \end{align*}.

This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of six.

\begin{align*}\frac{3x}{6} - \frac{2x}{6} = 6\end{align*}

Then we subtract the fractions to get \begin{align*}\frac{x}{6} = 6\end{align*}.

Finally we multiply both sides by 6 to get \begin{align*}x = 36\end{align*}.

Example

Example 1

Solve \begin{align*}\frac{2x}{5} - \frac{3x}{2} = 11 \end{align*}.

This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of ten.

\begin{align*}\frac{4x}{10} - \frac{15x}{10} = 11\end{align*}

Then we subtract the fractions to get \begin{align*}-\frac{11x}{10} = 11\end{align*}.

Finally we multiply both sides by \begin{align*}-\frac{10}{11} :\end{align*}

\begin{align*}-\frac{11x}{10}\cdot -\frac{10}{11} = 11 \cdot -\frac{10}{11} \end{align*}

to get \begin{align*}x = -10\end{align*}.

Review

Solve the following equations for the unknown variable.

1. \begin{align*}1.3x - 0.7x = 12\end{align*}
2. \begin{align*}-10a - 2(a+5) = 14\end{align*}
3. \begin{align*}5(2y-3y) = -20\end{align*}
4. \begin{align*}\frac{2}{3}x - \frac{1}{5}x = \frac{14}{15}\end{align*}
5. \begin{align*}5x - (3x + 2) = 1 \end{align*}
6. \begin{align*}s - \frac{3s}{8} = \frac{5}{6}\end{align*}
7. \begin{align*}10(y + 5y) = 10\end{align*}
8. \begin{align*}2.3x+2(0.75x-3.5) = 7.5\end{align*}
9. \begin{align*}3(x+2)+5(2-x)=-32\end{align*}
10. \begin{align*}6x + 2(5x - 2) = 12 \end{align*}

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