Algebra and Probability Study Guide 1 for McGraw-Hill's ASVAB
Practice questions for this study guide can be found at:
The Language of Algebra
Algebra uses arithmetic functions and processes, but some of the numbers are replaced by letters. The letters merely represent numbers that are currently unknown or that can change in value according to circumstances. In algebra, a letter representing a number that can change in value is called a variable.
- An expression such as 6x means "6 times some number, currently unknown" or "some number, currently unknown, times 6."
- An expression such as x + 7 means "some number, currently unknown, plus 7."
- An expression such as x – 12 means "some number, currently unknown, less 12."
- An expression such as means "some number, currently unknown, divided by 5," or "the ratio of some number and 5."
Very often verbal expressions in word problems need to be translated into algebraic expressions before they can be solved. Here are some examples of verbal expressions and their algebraic counterparts.
To evaluate an algebraic expression, substitute the given value for the unknown and then perform the arithmetic as indicated.
Evaluate a + b + c if a = 2, b = 4, and c = 3.
Substitute each value for the corresponding letter and then do the addition as indicated.
- 2 + 4 + 3 = 9
Solving Equations for One Unknown
An equation is a mathematical statement that contains an equal (=) sign. When an equation contains a letter standing for an unknown number, you can use the equation to find the value of that unknown. This is called solving the equation for the unknown.
Think of an equation as a balanced scale. Everything to the right of the = sign has to balance with everything on the left side of the = sign.
Because an equation is balanced, it will stay in balance if you do the same thing to the numbers on both sides of the = sign. For example, the equation 10 = 10 will stay balanced if you add 3 to both sides. The new equation will be 13 = 13. Similarly, the equation x + y = x + y will stay balanced if you subtract 10 from both sides. The new equation will be x + y – 10 = x + y – 10.
Similarly, the equation x + y = a + b will stay balanced if you subtract 8 from both sides. The new equation will be x + y – 8 = a + – – 8.
To solve an equation and find the value of an unknown, you need to get the unknown on one side of the equation and all the other terms on the other side of the equation. Consider this example.
Solve: y – 4 = 20
- Add 4 to both sides:
- y – 4 + 4 = 20 + 4
- y = 20 + 4
- y = 24
An equation will also stay balanced if you multiply or divide both sides by the same number. So you can also use these operations to solve equations.
Solve: 3x = 18
You want to get x all alone on the left side of the equation, so divide 3x by 3. Since 3/3 = 1, 3x/3 = x. To maintain the balance, divide the right side of the equation by 3 as well: = 6. So x = 6.
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