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(Video) Solving Linear Equations Using Reciprocals

by mathman1024

Here's a video by mathman1024 shows you how to solve equations using reciprocals.

$\large{\text{Summary}}$

A reciprocal of a number is $1$ divided by the number. In other words, you flip the number. So if you have $\frac23$, the reciprocal is $\frac32$. We flipped the numerator and denominator. For whole numbers, like $3$, we can imagine a $1$ being in the denominator so when we flip it to find the reciprocal we get $\frac13$.

A number multiplied by it's reciprocal always equals $1$. $$\frac{a}{b}\cdot\frac{b}a=1$$ We can use this along with the multiplication property to solve equations. The first one he looks at is: $\frac23x=20$. $$\begin{align} &\frac23x=20\\ &\color{#29A603}{\frac32\bigg(}\frac23x\color{#29A603}{\bigg)}=\color{#29A603}{\frac32(}20\color{#29A603}{)}\\ &x=30 \end{align}$$

Because we wanted to get the variable by itself, we multiplied the fraction by its reciprocal because that would make it a $1$ which we can ignore. But as always, we multiplied one side by something so we also have to multiply the other side by the same thing.

Try the other problem by yourself and see if you get the same answer.

1. $-\frac{14}9y=\frac{56}{81}$

If you've got a variable that's part of a fraction, you've got some options on how you want to handle it. Because fractions can be a pain in the butt, it's helpful to have multiple tools in your anti-fraction arsenal to handle them when they arise.

One tool is multiplying both sides by the fraction's denominator. A second tool uses something called reciprocals.

You might hear reciprocals called "inverses". There's also the much less technical description of reciprocals as "fractions flipped upside down".

If there is a negative sign on the fraction and you want the fraction's reciprocal, leave the negative sign where it is. Negative signs don't go away when you find a reciprocal.

This means that when you multiply a fraction by its reciprocal, they both disappear from the equation (since their value is now $1$). For example:

$$\begin{gather} \frac{4}{3} \cdot \frac{3}{4}\\ \frac{4\cdot3}{3\cdot4}\\ \frac{12}{12}\\ 1 \end{gather}$$

Let's look at two slightly more complex examples of reciprocals.

When you've got a fraction with a negative sign, the reciprocal is also negative. Remember, in order to check if two fractions are reciprocals, when you multiply them together, you should get $1$. In this case, the reciprocal is:

$$-\frac{7}{6}$$

If we had decided to leave the negative sign off the reciprocal, multiplying the two fractions together would give us $-1$. We don't want that. In order for a fraction to truly "disappear" from an equation, the product of it and its reciprocal needs to equal $1$. So, if there's a negative sign in the fraction, there's a negative sign in its reciprocal.

Yes, even whole numbers have reciprocals. The key is to re-write the number as a fraction. We can re-write $5$ as $\frac{5}{1}$. This should help you find the reciprocal, which turns out to be $\frac{1}{5}$. You can check your work here by multiplying your original value by the reciprocal, and making sure the product is $1$.

Now that we've got the idea of reciprocals straightened out, let's work on a few examples of equations that you can use reciprocals to solve.

Using reciprocals to solve one-step equations

$$\frac{2}{5}x = 6$$

Before we use the reciprocal to solve this problem, let's do a quick throwback to using plain old multiplication. Note that this equation can be re-written as

$$\frac{2x}{5} = 6$$

If you wanted, you could easily multiply both sides of the equation by the denominator ($5$) and then solve for $x$. That is a perfectly acceptable way of solving this equation.

Let's now consider the equally good approach of using reciprocals.

The fraction in our problem is $\frac{2}{5}$. This means that the reciprocal is $\frac{5}{2}$.

To get rid of our fraction, we need to multiply it by its reciprocal. And remember that whatever you do to one side of the equation, you have to do to the other.

$$\begin{align} \frac{2}{5}x &= 6\\ \\ \frac{2}{5}x \times \frac{5}{2} &= 6 \times \frac{5}{2}\\ 1x &= \frac{30}{2}\\ x &= 15 \end{align}$$

(Remember that the $6$ in our equation can also be written as $\frac{6}{1}$).

Pretty cool, right? Using the reciprocal immediately got rid of the fraction for us, making the equation much easier to solve. Let's look at another example.

Using reciprocals to solve two-step equations

$$\frac{2}{3}x - 1 = 5$$

Just like using plain ol' multiplication, when we use reciprocals to solve an equation, we want to simplify it as much as we can beforehand. This means that we need to carry out operations like addition and subtraction included in the equation. Then we can multiply by the reciprocal and solve for $x$.

First, we move $-1$ over to the other side of the equation:

$$\begin{align} \frac{2}{3}x - 1 &= 5 \\ \frac{2}{3}x - 1 + 1 &= 5 + 1\\ \frac{2}{3}x &= 6 \end{align}$$

Then, we can use the reciprocal to get $x$ alone on one side:

$$\begin{align} \frac{2}{3}x &= 6\\ \left(\frac{3}{2}\right) \times \frac{2}{3}x &= 6 \times \left(\frac{3}{2}\right) \\ x &= \frac{18}{2}\\ x &= 9 \end{align}$$

Let's try one more example:

$$\frac{5}{3}(x - 1) = 10$$

In this equation, the fraction applies to the whole quantity in parentheses. While you could distribute the $\frac{5}{3}$ to both $x$ and $-1$, it is actually simpler to multiply both sides by the reciprocal first.

First, we'll multiply by the reciprocal this time:

$$\begin{align} \frac{5}{3}(x - 1) &= 10\\ \\ \left(\frac{3}{5}\right) \times \frac{5}{3}(x - 1) &= 10 \times \left(\frac{3}{5}\right)\\ 1(x - 1) &= \frac{30}{5}\\ x - 1 &= 6 \end{align}$$

Now, we can just add $1$ to both sides to find the answer!

$$\begin{align} x - 1 &= 6\\ x - 1 + 1 &= 6 + 1\\ x &= 7 \end{align}$$

A reciprocal is a fraction flipped. So if your fraction is $\frac{a}{b}$ then the reciprocal is $\frac{b}{a}$ . A fraction times its reciprocal is always equal to 1.

For whole numbers like $1$, $2$, or $3$ just think of them in the form $\frac{1}{1}, \frac{2}{1},$ or $\frac{3}{1}$ before you flip them. (Their reciprocals would be $\frac{1}{1}, \frac{1}{2},$ and $\frac{1}{3}$ respectively).

So when you have an equation in the form $\frac{a}{b}x=c$. To isolate the variable $x$ all you do is multiply both sides by the reciprocal of $\frac{a}{b}$ because this will give you $x$ by itself. This reciprocal is $\frac{b}{a}$ and is the multiplicative inverse of $\frac{a}{b}$.

Example

The plan is to keep the $x$ on the same side of the equal sign. We identify the reciprocal of $\frac{4}{3}$, and multiply both sides of the equation by that number.

$$\begin{align} \frac{4}{3}x&=12\\ \\ \frac{4}{3}x &= 12\\ \frac{3}{4} \times\frac{4}{3}x &= 12\times\frac{3}{4}\\ \frac{12}{12}x &=\frac{36}{4}\\ 1x &= 9\\ x &= 9 \end{align}$$

Let's look at another example modeled below.

Example

$$\frac{5x}{6}=3$$

So, we start with $\frac{5x}{6}=3$ and want to multiply by the reciprocal $\frac{6}{5}$. We see,

$$\begin{align} \left(\frac{5x}{6}\right)&=3\\ \left(\frac{6}{5}\right)\cdot{\left(\frac{5x}{6}\right)}&=3\cdot{\left(\frac{6}{5}\right)}\\ x&=\frac{18}{5} \end{align}$$

As a side note, you could also write this improper fraction as a mixed number, $3 \frac{3}{5}$. We call this converting from an improper fraction to a mixed number.