Algebra Review for Armed Services Vocational Aptitude Battery (ASVAB) Study Guide (page 3)
Algebra questions do not appear on every test. However, when they do, they typically cover the material you learned in pre-algebra or in the first few months of your high school algebra course. Popular topics for algebra questions include:
- solving equations
- positive and negative numbers
- algebraic expressions
What Is Algebra?
Algebra is a way to express and solve problems using numbers and symbols. These symbols, called unknowns or variables, are letters of the alphabet that are used to represent numbers.
For example, let's say you are asked to find out what number, when added to 3, gives you a total of 5. Using algebra, you could express the problem as x + 3 = 5. The variable x represents the number you are trying to find.
Here's another example, but this one uses only variables. To find the distance traveled, multiply the rate of travel (speed) by the amount of time traveled: d = r × t. The variable d stands for distance, r stands for rate, and t stands for time.
In algebra, the variables may take on different values. In other words, they vary, and that's why they're called variables.
Algebra uses the same operations as arithmetic: addition, subtraction, multiplication, and division. In arithmetic, we might say 3 + 4 = 7, while in algebra we would talk about two numbers whose values we don't know that add up to 7, or x + y = 7. Here's how each operation translates to algebra:
An equation is a mathematical sentence stating that two quantities are equal. For example:
2x = 10
x + 5 = 8
The idea is to find a replacement for the unknown that will make the sentence true. That's called solving the equation. Thus, in the first example, x = 5 because 2 × 5 = 10. In the second example, x = 3 because 3 + 5 = 8.
Sometimes you can solve an equation by inspection, as with the above examples. Other equations may be more complicated and require a step-by-step solution, for example:
The general approach is to consider an equation like a balance scale, with both sides equally balanced. Essentially, whatever you do to one side, you must also do to the other side to maintain the balance. Thus, if you were to add 2 to the left side, you would also have to add 2 to the right side.
Let's apply this balance concept to our complicated equation above. Remembering that we want to solve it for n, we must somehow rearrange it so the n is isolated on one side of the equation. Its value will then be on the other side. Looking at the equation, you can see that n has been increased by 2 and then divided by 4 and ultimately added to 1. Therefore, we will undo these operations to isolate n.
- Begin by subtracting 1 from both sides of the equation:
- Next, multiply both sides by 4:
- Finally, subtract 2 from both sides:
- Which isolates n and solves the equation: n = 6.
Notice that each operation in the original equation was undone by using the inverse operation. That is, addition was undone by subtraction, and division was undone by multiplication. In general, each operation can be undone by its inverse.
After you solve an equation, check your work by plugging the answer back into the original equation to make sure it balances. Let's see what happens when we plug 6 in for n:
Solve each equation:
- x + 5 = 12
- 3x + 6 = 18
Positive and Negative Numbers
Positive and negative numbers, also known as signed numbers, are best shown as points along the number line:
Numbers to the left of 0 are negative and those to the right are positive. Zero is neither negative nor positive. If a number is written without a sign, it is assumed to be positive. Notice that when you are on the negative side of the number line, numbers with bigger values are actually smaller. For example, –5 is less than –2. You come into contact with negative numbers more often than you might think; for example, very cold temperatures are recorded as negative numbers.
As you move to the right along the number line, the numbers get larger. Mathematically, to indicate that one number, say 4, is greater than another number, say –2, the greater than sign (>) is used:
4 > –2
On the other hand, to say that –2 is less than 4, we use the less than sign, (<):
–2 < 4
Arithmetic with Positive and Negative Numbers
The table below illustrates the rules for doing arithmetic with signed numbers. Notice that when a negative number follows an operation (as it does in the second example below), it is enclosed in parentheses to avoid confusion.
When more than one arithmetic operation appears, you must know the correct sequence in which to perform the operations. For example, do you know what to do first to calculate 2 + 3 × 4? You're right if you said, "multiply first." The correct answer is 14. If you add first, you will get the wrong answer of 20. The correct sequence of operations is:
Even when signed numbers appear in an equation, the step-by-step solution works exactly as it does for positive numbers. You just have to remember the arithmetic rules for negative numbers. For example, let's solve 14x + 2 = 5.
- Subtract 2 from both sides:
- Divide both sides by –14: –14x ÷ –14 = –7 ÷ –14
Now try these problems with signed numbers.
- 1 – 3 × (–4) = x
- –3x + 6 = –18
- + 3 = –7
An algebraic expression is a group of numbers, unknowns, and arithmetic operations, like: 3x – 2y. This one may be translated as, "3 times some number minus 2 times another number." To evaluate an algebraic expression, replace each variable with its value. For example, if x = 5 and y = 4, we would evaluate 3x – 2y as follows:
3(5) – 2(4) = 15 – 8 = 7
Evaluate these expressions.
- 6a + 2b; a = 2 and b = –1
- 3mn – 4m + 2n; m = 3 and n = –3
- –2x – y + 4z; x = 5, y = –4, and z = 6
- The volume of a cylinder is given by the formula V=πr2h, where r is the radius of the base and h is the height of the cylinder. What is the volume of a cylinder with a base radius of 3 and height of 4? (Leave π in your answer.)
- If x = 3, what is the value of 3x – x?
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