Learn Equations: Easy Equations with Rational Numbers
Solving equations with rational numbers is one of the first skills that you'll learn in algebra. In fact, you may have already learned to solve equations without realizing it! Read on to learn the basics of solving equations with rational numbers.
Simple Rational Number Equations
In math, an equation is a statement that tells you two expressions are equal. An expression can be a single number, like 12, or it can contain numbers, variables and operation signs, like this: 3(x + 7)/4. Most of the equations you'll see in algebra will have rational numbers, which are numbers that can be written as fractions. They include integers and fractions, as well as most square roots and decimals.
Another important fact about equations is that they can be 'true' or 'false.' For instance, the equation 5 = 5 is true, since five does equal five. However, the equation 5 = 7 is false, since five does not equal seven. When you're solving an equation, you have to assume that it is true.
The Property of Equality
To solve equations, it's important to understand this rule: if you perform an operation on one side of an equation, you must perform the same operation on the other side for the equation to remain true. Here's an example with a very simple equation:
5 = 5
5 + 7 = 5 + 7
12 = 12
In this example, we performed the same operation - adding seven - on each side of the equation, so the equation remained true. If we had only added seven to one side of the equation, it would have stated that 12 = 5, which is not true. Keep this rule in mind as you read through the next section about solving equations.
How to Solve Equations
Solving an equation is just like solving a mystery. You are the detective, and your first clue is the 'equals' sign in the equation. It tells you that the expressions to the left and right of it are equal. To crack the case of the unknown number, you must figure out what number will make the equation true when it is substituted for the variable.
There are two approaches you can take to uncover the identity of the missing number. One is to solve the equation in your head, using mental math. For example, you can look at the equation 4 + x = 9, and mentally calculate that 4 + 5 = 9, so x = 5. But what do you do if your equation has larger numbers, and you can't figure it out in your head? For instance, how do you solve an equation like 4,378 + x = 10,524?
First, think back to how you solved the equation 4 + x = 9 in your head. You might have asked yourself, 'What number plus four equals nine?' You may also have wondered, 'Nine minus what number equals four?' The answer to both of these questions is five, of course. Although you may not have realize it, here's how you solved that equation:
4 + x = 9
4 - 4 + x = 9 - 4
x = 5
You subtracted four from both sides of the equation. On the left side, this meant that you subtracted four from four, leaving you with 0 + x, or just x. On the right side, you subtracted four from nine, leaving you with just five. As long as you perform the same operation on both sides of the equation, the equation will still be true. You can use the same strategy for equations with larger numbers:
4,378 + x = 10,524
4,378 - 4,378 + x = 10,524 - 4,378
x = 6,146
Tip: You'll probably hear your math teacher talking about 'isolating the variable,' which is just a fancy way to say 'getting the variable on one side of the equals sign all by itself.' This is what we did in the two examples above. It is an essential step in solving the equation, because once the variable is by itself, you know that it is equal to the expression on the other side of the equation.
For problems one and two, you'll use the same steps from the examples above. For problems three through five, you'll need to use your detective skills. Just remember that you must perform the same operation on both sides of the equation, and that you want to pick an operation that will help you get the variable by itself.
1. 8 + x = 15
2. 745 + y = 1,000
3. y - 12 = 4
4. x - 438 = 232
5. x ÷ 2 = 4
1. To solve 8 + x = 15, you need to find the value of x. To do this, subtract eight from both sides of the equation to get the solution, x = 7.
8 + x = 15
8 - 8 + x = 15 - 8
x = 7
2. Here's the solution for this equation:
745 + y = 1,000
745 - 745 + y = 1,000 - 745
y = 255
3. You'll need to use a slightly different approach to solve this equation. Since the 12 is negative, you'll need to add 12 to both sides of the equation to get the variable, y, by itself.
y - 12 = 4
y - 12 + 12 = 4 + 12
y = 16
4. Although x - 438 = 232 has bigger numbers, you'll still use the same process to solve it that you used for the previous problem:
x - 438 = 232
x - 438 + 438 = 232 + 438
x = 670
5. To solve x ÷ 2 = 4, you need to get the x by itself. You may have figured out that to accomplish this, you need to use the inverse, or 'opposite,' operation. In problems one and two, when a number was being added to the variable, you subtracted that number from both sides, since subtraction is the inverse of addition. In problems three and four, where a number was subtracted from the variable, you added it to both sides.
In this problem, the variable is being divided by a number. The inverse of division is multiplication, so you need to multiply both sides of the equation by the number, like this:
x ÷ 2 = 4
2(x ÷ 2) = 2(4)
x = 8
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