In algebra you work with the four operations of addition, subtraction, multiplication, and division. The chart below shows the symbols that are used for these four operations.
| OPERATION | SYMBOL | EXAMPLE | MEANING |
|---|---|---|---|
| Addition | + | 5+n | 5 plus n |
| Subtraction | - | 8-n | 8 take away n |
| Multiplication | • | 5•3 | 5 times 3 |
| Multiplication | () | 5(3) | 5 times 3 |
| Multiplication | no symbol | 5n | 5 times n |
| Division | ÷ | n÷3 | n divided by 3 |
Whenever a number and a variable are multiplied together, the number part is called the coefficient of the variable. In the expression 7x, the coefficient of x is 7. The variable is multiplied by 7.
TIP:
Whenever the variable is written without a number in front of it, the coefficient is understood to be 1. Therefore, x means 1x, and the coefficient is 1.
Algebraic Expressions
An algebraic term can be a constant, a variable, or the multiplication or division of numbers with variables. Some examples of terms are 4, y, 3n.
An algebraic expression is any combination of numbers, variables, grouping symbols, and operations. In an algebraic expression, terms are separated by + and - signs. The sign before the term belongs to the term. For example, 5 - 3y, a + 4, and m/4 + 2 are all algebraic expressions.
5 - 3y is an algebraic expression that has two terms: 5 is first term and -3y is second term.
a + 4 is an algebraic expression that has two terms: a is first term and +4 is second term.
m/4 + 2 is an algebraic expression that has two terms: m/4 is first term and +2 is second term.
Evaluating Algebraic Expressions
An algebraic expression has no value until the variables in the expression have been replaced with numbers. To evaluate an expression, replace every variable with a given value for the variable. Then evaluate the resulting numerical expression. Remember to follow the rules for the order of operations and the rules for signed numbers.
Evaluate the algebraic expressions for the given values of the variables.
Example 1 Evaluate 3x - 4 when x is 5.
3x - 4 = 3(5) - 4 = 15 - 4 = 11
Example 2 Evaluate 7 + 4(8 - x) when x is -2.
7 + 4(8 - x) = 7 + 4(8 - (-2)) = 7 + 4(8 + 2) = 7 + 4(10) = 7 + 40 = 47
In Example 2 you must do the work inside the parentheses first. 8 - (-2) means 8 + 2 following the rules for signed numbers.
Equations
An equation states that one expression is equal to another expression. For example, the equation x + 7 = 10 sets the expression x + 7 equal to the expression 10. In the equation x + 3 = 2x - 9, the expression x + 3 is set equal to the expression 2x - 9.
It is important to recognize the difference between an expression and an equation.
An equation always has an equal sign.
Expression Equation
x - 7 x + 7 = 10
2x - 9 x + 3 = 2x - 9
AN EQUATION ALWAYS HAS THREE PARTS
LEFT-SIDE EXPRESSION EQUAL SIGN RIGHT-SIDE EXPRESSION
An equation may be true or false depending on the replacement value for the variables. The equation x + 7 = 10 says that some number x added to 7 equals 10. Since you know that 3 + 7 = 10, you know that x = 3 is the solution of the equation. The solution is the value of the variable that makes the statement true. If you had said that x = 2, you would not have found the solution of the equation. You solve an equation when you find the solution for the variable. On the GED Mathematics Test, you may have to translate a word problem into an equation. You will also write and interpret your own equations.
Example Solve the equation 2x + 5 = 17.
This equation means that a number multiplied by 2 and added to 5 equals 17.
Do you know what the number is? You are right if you said 6, because
2(6) + 5 = 17.
Solving One-Step Equations
Among the most important uses of algebra are solving equations and using equations to solve word problems.
Eyeballing the Solution
Solving an equation means finding the value of the variable that makes the equation
true. Some equations will be fairly easy to solve because you know your math facts.
If you can look at the equation and see the answer, you are "eyeballing" the equation.
For example, x - 4 = 10 is an equation. What value of x will make this true? You would choose x = 14 because you know 14 - 4 = 10. Another example is 3x = 15. What value of x makes this equation true? The answer is 5 because you know 3(5) = 15.
Algebraic Solutions
Let's look at algebraic methods of solving one-step equations. Before you start solving equations, you must understand a couple of mathematical ideas.
First, an equation is a perfect balance between what is on the left side of the equal sign and what is on the right side of the equal sign. If you make any changes on the left side, you must make the same changes on the right side. For instance, if you add 7 to the left side, you must add 7 to the right side for the two sides to remain equal. Then you know the result will be true.
Second, your goal in solving an equation is to get the variable all by itself on one side of the equation. Concentrate on the variable. In the equation x + 3 = 10, concentrate on the x. Notice that the x is being added to 3. This will tell you how to solve the equation.
TIP:
Keep the equation in balance and concentrate on the variable.
To solve an equation, perform the opposite operation (called the inverse operation). Addition and subtraction are opposites, and multiplication and division are opposites. In the equation x + 3 = 10, you see that 3 is being added to x. The opposite of adding three is subtracting three. Subtract three from both sides to get x all by itself.
Example x + 3 = 10
Step 1 Look at x. 3 is added to it. x + 3 = 10
Step 2 Subtract 3 from both sides. x + 3 - 3 = 10 - 3
Step 3 Solve the equation. x = 7
SOLVING AN EQUATION
1. Keep the equation in balance.
2. Concentrate on the variable.
3. Perform the opposite operations.
4. Isolate the variable on one side of the equation.
No comments:
Post a Comment