Equações com dois termos no numerador

When you see two terms in the numerator of a fraction, the first thing to remember is: don't panic. Although having more than one term can seem like a lot to deal with, most of the time, it's a pretty easy fix.

The key is to get rid of the fraction. Now, "get rid of" does not mean "erase it because no one likes fractions." What it does mean is that we need to multiply both sides of the equation by the denominator (the bottom part of the fraction), so the fraction goes away.

Take a look at this equation:

$$\frac{2x+8}{2} = 6$$

We've got two terms in the numerator: there's the $2x$ and the $8$.

So, let's get rid of the fraction by multiplying both sides of the equation by the denominator. Because a fraction is basically division, we want to do the opposite sign: multiplication! $$2\left(\frac{2x+8}{2}\right) = 2(6)\\ 2x + 8 = 12$$

POOF! The fraction is gone! It's almost like it's magic...but it's math! Now we have a much more manageable and familiar equation to work with. The two equations are still equivalent.

Remember: whatever you do on one side of the equation, you have to do on the other. Follow the Addition and Subtraction Property of Equality and the Multiplication and Division Property of Equality!

That wasn't so bad, right?

To double check the problem's solution, plug in the value you solved for $x$, which in this case is $2$.

But what if you come across a problem where there's two terms in the numerator and two terms in the denominator? (Math teachers can be so crafty!)

Not to worry—the two rules above will still work!

1. Don't panic.
2. Get rid of the fraction by multiplying both sides of the equation by the denominator.

Let's practice a little.

$$\frac{3x - 1}{2x + 8} = 2$$

Get rid of the fraction by multiplying both sides of the equation by the denominator. Remember, multiply both sides by the whole denominator.

Now you have variables on both sides of the equation!

$$\begin{align} (2x + 8) \left(\frac{3x - 1}{2x + 8}\right) = 2 (2x + 8)\\ 3x - 1 &= 4x + 16 \end{align}$$

Next, solve the equation. Remember to combine like terms and get anything with an $x$ on one side of the equation all by itself. Always remember: whatever you do on one side of the equation, you have to do on the other.

$$\begin{align} 3x - 1 &= 4x + 16\\ 3x - 1 + 1 &= 4x + 16 + 1\\ 3x &= 4x + 17\\ 3x - 4x &= 4x + 17 - 4x\\ -x &= 17 \\ x &= -17 \end{align}$$

(Remember, we want $x$ to be alone on one side of the equation and we want it to be positive. To make our $-x$ positive, we need to divide each side of the equation by $-1$, giving us a final answer of $x=-17$).

So, the important things to remember when facing a fraction with two terms in the numerator are:

1. Don't panic. There's a way to fix this.
2. Get rid of the fraction by multiplying both sides of the equation by the denominator.
3. Solve for $x$ as usual.

(Video) Solve for x in Two Steps With Fractions (Simplifying Math)

by Eric Buffington

In the video above, Mr. Buffington walks through a couple examples of solving equations with two terms in the numerator.

$\large{\text{Summary}}$

He tells us that there are $3$ steps to solving these kind of problems.

1. Find the variable.
2. What happened? What is the variable connected to?
3. Do the inverse to both sides.

We see, $$\begin{align} \frac{x-2}{\color{#8925AE}{(-6)}}&=12\\ {\color{#8925AE}{(-6)}}\frac{x-2}{-6}&=12\cdot{\color{#8925AE}{(-6)}}\\ x\,\color{#29A603}{-\,2}&=-72\\ x-2\,\color{#29A603}{+\,2}&=-72\,\color{#29A603}{+\,2}\\ x&=-70 \end{align}$$

Watch the video for the other example. Now, look at the graph of $y=\frac{x-8}{-4}$.

An equation with two terms in the numerator can be solved by multiplying both sides by the denominator, and then solving like you would normally solve an equation (with division, subtraction, or using some combination of those tools). For instance: $$\begin{align} \frac{x-4}{2} &= 5 \text{ (Multiply both sides by $2$)}\\ 2 \times \frac{x-4}{2} &= 2 \times 5 \end{align}$$

When you multiply something by $2$ and divide it by $2$, the multiplication and division cancels out (e.g. $\frac{3⋅4}{3} = \frac{12}{3} = 4$) So we get: $$\begin{align} x-4 &= 10\quad \text{(Add 4 to each side)}\\ x - 4 + 4 &= 10 + 4\\ x &= 14 \end{align}$$

$$\begin{align} \frac{5+x}{2} &= 4 \\ \frac{5+x}{2} \times2 &= 4 \times 2 \\ 5 + x &= 8 \\ 5 + x - 5 &= 8 - 5 \\ x &= 3 \end{align}$$

Example: Two Terms in the Numerator

Having two terms in the numerator is similar to having just one term in the numerator. In both cases, we first want to get rid of the denominator.

Let's start with an equation: $$\frac{2x+5}{5}=6$$ To start:

To "get rid of" the $5$, we multiply both sides of the equation by $5$.

Then, we solve for $x$ with some more manipulation of the equation.

Remember, always manipulate both sides of the equation at the same time. If you do something to one side of the equation, make sure you also do that same thing to the other side!