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Tuesday 21st of November
 
   
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Algebra Expressions
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Slopes of Perpendicular Lines
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Roots - Radicals 1
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Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
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Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Percents
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
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Polar Form of a Complex Number
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Algebra
Common Logs
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Multiplying Fractions in General
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Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
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The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
fractions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
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Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
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Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Quadratics
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Decimals
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
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Algebra Expressions

Why study algebra? Because this topic provides the mathematical tools for any problem more complicated than just combining some given numbers together.

Algebra lets you solve word problems in a regular and systematic way. Algebra le ts you use symbols to solve all possible instances of a certain equation, not just a single example of the equation with certain numbers in it. For example, it will help you understand the properties of the universal equation for a straight line.

Really, What is Algebra?

Practically speaking, algebra has two main differences from the arithmetic you’ve studied so far:

1. We use variables as placeholders for unknown quantities.

2. We work on whole equations rather than just numbers.

What we’re teaching in algebra is “how you can torture equations” to get the information you want. We will happen to do some arithmetic along the way, but the main idea is to learn how to change and transform equations. Sometimes the information you want is a simple number for a variable, and sometimes you want to know how one variable is related to another.

 

Using Variables

It is common to have problems in which something is unknown. For example:

3 × 4 = _____ or 7 + _____ = 10

 In algebra, we change from using blanks or question marks to using letters or variables. (Of course, the blanks above may be correctly filled in using 12 and 3, respectively!) Using 'x' to stand for the first blank number and 'y' for the second we could write:

3 × 4 = x and 7 + y = 10

These are really the same problems as the fill- in-the-blank problems above. In this case, we could say the solution is x = 12 in the first equation and y = 3 in the second.

The letters are variables that are really just unknown numbers as was the blank. We can use variables in any way that we could use other numbers. We can add and subtract, multiply and even use them in powers and square roots.

There isn't anything special about 'x' and 'y', either. Although they are the most common letters used, we can choose any letters we want. We will often choose letters that remind us of the missing number. For instance, we might use h for height, l for length, v for volume and t for time.

Letters are not reserved, and you really can choose what ever you like. But there are some commonly used ones. The letters a, b and c are often used to indicate a constant. They indicate a number of unknown value which is not variable; it is an unchanging number. We just don’t know what number it is.

 

A Short Form for Multiplication

Wait a minute! There could be some confusion with the letter x. Does it mean the unknown number ‘x' or to multiply? To avoid confusion, we have some new ways to indicate multiplication.

We can write the two multiplicands next to each other. If multiplication is not obvious, we can use a dot • as a separator.

In fact, we will rarely use the symbol “×” anymore:

· Use a dot • between two numbers:  4•7 “four times seven”
· Or just write a number by a variable: 8t “eight times t”
· Write a value next to parentheses:  4(x + 1) “four times x plus one”
or: 2(3) “two times three”
· Write a value next to an operation: “4 times the square root of 25”
· Write to operations next to each other: 3!4! “3 factorial times 4 factorial”
· Write parentheses next to each other: (x + 1)(x - 1) “x plus 1 times x minus 1”

Neatness counts! You must make the dot • clearly different than a decimal point. Be sure the reader can tell when you mean 4•7 instead of 4.7!

 

Expressions

Expressions are numbers combined together using various arithmetic operations.

For example, the expression 4 • 9 / 6 combines the numbers 4, 9 and 6 by multiplying 4 by 9 and then dividing by 6 giving the result 6.

Sometimes an expression will involve a variable. You can simplify the expression by substituting the value of the variable into the expression and compute the result:

Example:

Find 3x - 6 if you are given that x = 7

Solution:

Put the value of 7 in for x

Multiply by 3 to get 21

Subtract 6 to get 15

3 • 7 - 6

21 - 6

15

 

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