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Algebra

Algebra is a way of expressing mathematical equations using numbers, and letters and symbols called variables. For example, the equation below is used to show that the speed that something travels multiplied by the time that it travels will result in the distance traveled.

In standard algebra, a, b, c, and d tend to signify known variables while x, y, and z tend to signify unknown variables. Also note that a common algebraic shortcut is to eliminate the multiplication sign. Two letters next to each other indicate that they are multiplied together.

Algebraic properties

There are several standard algebraic properties. The first property is the commutative property. It is important to note that this property only applies to multiplication and addition and not division and subtraction. This property also is only valid if all the terms are multiplied or if all the terms are added. It does not apply to a mix of addition and multiplication. This property states that order does not matter. Reversing the terms does not affect the outcome.

The second property is the associative property. This property also only considers multiplication and addition and only applies if all the terms are multiplied or if all the terms are added. This property deals with when more than two terms are present and states that it does not matter which terms are multiplied or added first.

The next property is the distributive property and deals with parentheses. It is important that unless parentheses are involved, multiplication is always performed first. If there are parentheses, the terms inside the parentheses are dealt with first. If a term is multiplied by some terms inside parentheses, that term is multiplied by each of the terms inside. If two sets of parentheses are multiplied together, each term inside the first set of parentheses are multiplied by each term inside the second set of parentheses. If there is a plus sign in front of the parentheses, all of the terms inside stay the same sign. If there is a negative sign in front of the parentheses, all of the terms switch signs.

Units

When using measurements, units are very important. It is important that all of your units are in the same standard whether they are English measurements or Metric units. If speed is measured in miles per hour, time should be in hours and distance should be in miles.

Equations

Equations in algebra are statements that two expressions are equal. As long as the same thing is done to each side of the equation, each side will remain equal. For instance, it is possible to add, subtract, multiply or divide terms to each side of the equation. This makes it possible to solve for unknown variables.

There are five rules to solving simple algebraic equations

  1. Clear any fractions by multiplying all the terms on both sides of the equation by the denominators
  2. Remove all parentheses
  3. Transpose all of the terms containing the unknown variable to one side of the equation and all known numbers or variables on the other side of the equation
  4. Combine like terms on each side of the equation
  5. Divide both sides by whatever coefficient is in front of the unknown variable.

Clear fractions by multiplying both sides by 4b

Transpose by moving all terms with x to the left side and all terms without x to the right side

Combine like terms

Divide both sides by the coefficient in front of x

Problem Solving

Algebra can be used to solve word problems like the following:

"A man is 3 times as old as his son, but ten years ago he was 5 times the age of his son. What are their ages now?"

There is only one unknown variable because once the age of one is found, it can be used to calculate the age of the other. So let the son's age be x making the man's age 3x. Ten years ago, the man's age was (3x-10) and the son's age was (x-10). Since the man's age ten years ago was five times that of his son's age, the following equation is obtained.

Using the steps for solving equations, the equation created from the word problem can be solved

This gives the son's age as 20 years and the man's age as 60 years.