2.2 Solve equations using the division and multiplication properties  (Page 3/4)

Solve: $\frac{y}{\mathrm{-7}}=\mathrm{-14}.$

Solution

Here $y$ is divided by $\mathrm{-7}$ . We must multiply by $\mathrm{-7}$ to isolate $y$ .

Multiply both sides by $\mathrm{-7}$ .
Multiply.
Simplify.
Check: $\frac{y}{\mathrm{-7}}=\mathrm{-14}$
Substitute $y=98$ .
Divide.

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Solve: $\text{−}n=9.$

Solution

Remember $\mathrm{-}n$ is equivalent to $\mathrm{-1}n$ .
Divide both sides by $\mathrm{-1}$ .
Divide.
Notice that there are two other ways to solve $\mathrm{-}n=9$ . We can also solve this equation by multiplying both sides by $\mathrm{-1}$ and also by taking the opposite of both sides.
Check:
Substitute $n=\mathrm{-9}$ .
Simplify.

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Solve: $\frac{3}{4}x=12.$

Solution

Since the product of a number and its reciprocal is 1, our strategy will be to isolate $x$ by multiplying by the reciprocal of $\frac{3}{4}$ .

Multiply by the reciprocal of $\frac{3}{4}$ .
Reciprocals multiply to 1.
Multiply.
Notice that we could have divided both sides of the equation $\frac{3}{4}x=12$ by $\frac{3}{4}$ to isolate $x$ . While this would work, most people would find multiplying by the reciprocal easier.
Check:
Substitute $x=16$ .

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In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

Solve: $\frac{8}{15}=-\frac{4}{5}x.$

Solution

Multiply by the reciprocal of $-\frac{4}{5}$ .
Reciprocals multiply to 1.
Multiply.
Check:
Let $x=-\frac{2}{3}$ .

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Solve equations that require simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

Solve: $14-23=12y-4y-5y.$

Solution

Begin by simplifying each side of the equation.

Simplify each side.
Divide both sides by $3$ to isolate $y$ .
Divide.
Check:
Substitute $y=\mathrm{-3}$ .

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Solve: $\mathrm{-4}(a-3)-7=25.$

Solution

Here we will simplify each side of the equation by using the distributive property first.

Distribute.
Simplify.
Simplify.
Divide both sides by $\mathrm{-4}$ to isolate $a$ .
Divide.
Check:
Substitute $a=\mathrm{-5}$ .

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Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Translate to an equation and solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.