带小数方程式

(Video) Solving Decimal Equations

by mathman1024

Here's a video by mathman1024 showing you how to solve equations and expressions with decimals. There is a video with more examples below.

$\large{\text{Summary}}$

The solving process does not change once decimals are included. You don't do anything different from solving regular equations. Just be careful and make sure you don't make any mistakes with the algebra.

Let's look at an example: $$0.25x-1.7=4.9$$ Our goal is to isolate the variable. $$\begin{align} 0.25x-1.7&=4.9\\ 0.25x-1.7\,\color{#8925AE}{+\,1.7}&=4.9\,\color{#8925AE}{+\,1.7}\\ 0.25x&=6.6\\ \frac{0.25x}{\color{#8925AE}{0.25}}&=\frac{6.6}{\color{#8925AE}{0.25}}\\ x&=26.4 \end{align}$$

(Video) Solving Linear Equations with Decimals

by mathman1024

Here's another video with more examples. Try them out on your own first and then watch the video to see the answers!

1. $2.6x+11.4=1.1x-7.2$
2. $5x=1.21$

Equations with decimals are like the calmer, less annoying little cousin of equations with fractions. They're still something you've got to watch out for, but they're also likely to cause you a lot less brain-ache.

The approach to solving equations with decimals is pretty much the same as the approach with non-decimal numbers. The goal is to isolate terms with $x$ on one side of the equation. That said, decimals can look really different than whole numbers.

Let's start out with an equation that contains decimals: $$0.4(x + 3.6) = 2$$

In Approach 1, we multiplied both sides of the equation by $0.4$ first, then isolated $x$ on one side to solve. In Approach 2, we distributed $0.4$ and then use subtraction and division to solve.

Either of these approaches works and will get you to the same value for $x$. This is also a good way to double check your work!

We can also look at it this way:

Now let's try an equation that contains decimals and fractions. $$\frac{x}{4} + 2.25 = 3$$

Despite the way this equation looks, it's really not much different than the one we looked at earlier. As usual, we want to isolate the term with the $x$ on one side. $$\frac{x}{4} = 0.75$$ From here, solve for $x$.

$$\begin{align} \frac{x}{4} &= 0.75\\ 4\left(\frac{x}{4}\right) &= 4(0.75)\\ x &= 3 \end{align}$$

Let's take a look at one more type of equation with fractions and decimals. $$\frac{4}{5} + 0.38x = 2$$

Start by converting the fraction into a decimal. This will make it easier to combine with the $2$ on the other side of the equation. $$0.80 + 0.38x = 2$$ Now, solve for $x$ like we usually do by isolating it on one side of the equation.

$$\begin{align} 0.80 + 0.38x &= 2\\ 0.80\,{\color{#8925AE}{-\,0.80}}+0.38x&=2\,{\color{#8925AE}{-\,0.80}}\\ 0.38x &= 1.20\\ \frac{0.38x}{\color{#8925AE}{0.38}}&=\frac{1.20}{\color{#8925AE}{0.38}}\\ x &= 3.16 \end{align}$$

Solving equations with decimals isn't much more complicated than solving equations with whole numbers—decimals can just be finicky. You can use any of the methods from the sections on two-step equations, equations with variables on both sides, and solve linear equations by grouping.

It may also help to review basic functions with decimals, like adding decimals, subtracting decimals, multiplying decimals, and dividing decimals. These concepts will, obviously, show up a lot throughout this lesson.

Example

Find $x$ in $0.5x+8.6=12.48$.

Just like we do with all other equations, we first want to get the variables on one side and the constants on the other. $$\begin{align} 0.5x + 8.6 &= 12.48\\ 0.5x + 8.6\,\color{#8925AE}{-\,8.6} &= 12.48\,\color{#8925AE}{-\,8.6}\\ 0.5x &= 3.88\\ \end{align}$$

Once we have our variable on the left side, we can divide by the coefficient of the variable, which will just leave the variable. $$\begin{align} 0.5x &= 3.88\\ \frac{0.5x}{\color{#8925AE}{0.5}} &= \frac{3.88}{\color{#8925AE}{0.5}}\\ x &= 7.76 \end{align}$$