In this lesson we will be looking at how to solve equations, starting with the simplest equations. The Additive Inverse Property is also explained.
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Solving one-step equations with adding and subtracting:
Many one-step equation such as $x+3=10$ can be solved in your head without the need to follow any specific procedures. Still we need to learn the procedures to solve these problems. What we are actually doing is providing you with the tools to help solve complex problems such as: $6(3x + 5)= \frac{4}{5} x + 16$.
So lets try tacking $n+4.5=8$
How do we find the value of $n$?
We can think of an equation as a balance, if we want to keep the balance level then what ever we do to one side we need to the other.
This problem can solve this problem two different ways:
We could subtract 4.5 from both sides of the equation.
| $n+4.5=8$ | $n+4.5$ $-4.5$ $=8$$-4.5$ | $n=3.5$ |
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Or we could add the opposite of $4.5$ (which is ${^-4.5}$) from both sides of the equation.
| $n+4.5=8$ | $n+4.5+$ ${^-4.5}$ $=8+$${^-4.5}$ | $n=3.5$ |
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It does not matter which method we use, you will still get the same answers.
In the second situation we used the Additive Inverse Property.
Additive Inverse Property:
any number plus its opposite will equal 0.
| $a+{^-a}$ | $=$ | $0$ |
| ${^-a}+a$ | $=$ | $0$ |
A few more examples:
| Example 2 | $9.65=x-3.7$ |
| Rewrite the subtraction problem as an addition problem | $9.65=x$$+{^-3.7}$ |
| Add the opposite of ${^-3.7}$ to both sides | $9.65$$+3.7$$=x+{^-3.7}$$+3.7$ |
| Solution | $5.95=x$ |
| Example 3 | $y-{^-7}=4 \frac{1}{2}$ |
| Rewrite the subtraction problem as an addition problem | $y$ $+$ $7$ $=$ $4\frac{1}{2}$ |
| Add the opposite of $7$ to both sides | $y+7$ $+$ ${^-7}$$=$ $4\frac{1}{2}$$+{^-7}$ |
| Solution | $y={^-2}\frac{1}{2}$ |
Recap:
Whenever we are solving an equation our goal is to isolate the variable we are solving for, when doing this we need to keep the equation balanced by performing the same operations to both sides of the equation.
