How to simplify an algebraic expression: 10 steps (with pictures)

How to simplify an algebraic expression: 10 steps (with pictures)
How to simplify an algebraic expression: 10 steps (with pictures)
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An algebraic expression is a set of numbers, variables, and operators. Although it is impossible to solve an algebraic expression (because it does not contain a sign), =) it is possible to simplify it. It is however possible to solve an algebraic equation which is in fact a set of two algebraic expressions separated by an equal sign.

Steps

Method 1 of 2: Understand the basics

Solve an Algebraic Expression Step 1

Step 1. Know the difference between an algebraic expression and an algebraic equation

An algebraic expression is a set of numbers, variables, and operators. An algebraic expression does not contain an equal sign and cannot be solved. On the contrary, an algebraic equation can be solved and consists of two algebraic expressions (which can also be called members) separated by an equal sign. Here are some examples.

  • An algebraic expression: 4x + 2
  • An algebraic equation: 4x + 2 = 100
Solve an Algebraic Expression Step 2

Step 2. Learn how to group similar terms together

Grouping similar terms together simply means adding (or subtracting) terms of the same degree. This means that all the terms in x2 can be grouped together, all terms in x3 can be grouped together and that all constants (numbers that are not attached to variables), such as 8 or 5 for example, can be combined together. Here is an example:

  • 3x2 + 5 + 4x3 - x2 + 2x3 + 9 =
  • 3x2 - x2 + 4x3 + 2x3 + 5 + 9 =
  • 2x2 + 6x3 + 14
Solve an Algebraic Expression Step 3

Step 3. Learn how to factor a number

If you have to solve an algebraic equation, which means that there is one member on each side of the equal signal, it is possible to factor both members by a factor common to both members. Observe the coefficients of all the terms (the numbers in front of the variables or the constants) and see if you find a factor common to all these coefficients (a number that divides all the coefficients in your equation). If you can do this (or on the contrary, it is not possible to do it), you have managed to simplify your equation and you are on the right path to solving it. Here is an example:

  • 3x + 15 = 9x + 30

    You can see that in this example 3 divides all the coefficients in your equation. Factor your two members by 3 (which is the same as dividing the two members of your expression by 3) to simplify your equation

  • 3x / 3 + 15/3 = 9x / 3 + 30/3 =
  • x + 5 = 3x + 10
Solve an Algebraic Expression Step 4

Step 4. Know the operating priorities

The operating priorities (which can be summarized under the acronym PEMDAS) give the order in which the different mathematical operations must be performed. The order is as follows: Parentheses, Exponents, Multiplication, Division, Addition and Subtraction. Here is an example to illustrate the operating priorities to be respected during your calculations.

  • (3 + 5)2 x 10 + 4
  • First, perform the operation in parentheses (the P of PEMDAS)
  • = (8)2 x 10 + 4
  • Then perform the operation relating to exponent 2 (the E of PEMDAS).
  • = 64 x 10 + 4
  • Then perform the multiplication.
  • = 640 + 4
  • Finally, finish with the addition.
  • = 644
Solve an Algebraic Expression Step 5

Step 5. Learn how to isolate the variables

If you are looking to solve an algebraic equation, then your goal is to place the variable (often written x) on one side of the equal sign and the constants on the other side of the equal sign. The usual operations to isolate x on one side of the equation are multiplication, division, addition, subtraction, square root. There are of course still other operations that can be used, but the ones listed here are the most common. Once x is isolated, it is easy to find the solution to your equation. Here is an example:

  • 5x + 15 = 65 =
  • 5x / 5 + 15/5 = 65/5 =
  • x + 3 = 13 =
  • x = 10

Method 2 of 2: Solve an algebraic equation

Solve an Algebraic Expression Step 6

Step 1. Solve a simple linear algebraic equation

A linear algebraic expression only contains constants and first degree terms (no exponent or anything like that). To solve this type of equation, you just need to use addition, subtraction, multiplication or division to isolate the variable (often noted x) and then find the solution to your equation. Here is an example:

  • 4x + 16 = 25 -3x =
  • 4x = 25 -16 - 3x
  • 4x + 3x = 25 -16 =
  • 7x = 9
  • 7x / 7 = 9/7 =
  • x = 9/7
Solve an Algebraic Expression Step 7

Step 2. Solve an algebraic equation containing a second order term

If your equation contains a second order term (noted x² if x is your variable), then you must first isolate the variable on one side of the equal sign and then solve the equation by "removing" the square of the variable and constant. Here is how to do it:

  • 2x2 + 12 = 44

    Start by subtracting 12 from both sides of the equation

  • 2x2 + 12 -12 = 44 -12 =
  • 2x2 = 32

    Then divide each side of the equation by 2

  • 2x2/2 = 32/2 =
  • x2 = 16

    • Solve by taking the square root of each side of the equation. Indeed this will transform the x2 in x.
  • √x2 = √16 =
  • write the two solutions on your sheet: x = 4 or -4
Solve an Algebraic Expression Step 8

Step 3. Solve an algebraic equation containing fractions

If you are looking to solve an algebraic equation that has fractions in it, you will need to start by using a rule of three, then group similar terms together and isolate the variable. Here is how to do it:

  • (x + 3) / 6 = 2/3

    First, start by using the rule of three to make fractions disappear. You must therefore multiply each member by the denominator of the fraction of the other member

  • (x + 3) x 3 = 2 x 6 =
  • 3x + 9 = 12

    Now group similar terms together. Group the constants, 9 and 12, by subtracting 9 from each side of the equation

  • 3x + 9 - 9 = 12 - 9 =
  • 3x = 3

    Isolate the variable, x, by dividing both sides of the equation by 3, and voila, you have your solution

  • 3x / 3 = 3/3 =
  • x = 3
Solve an Algebraic Expression Step 9

Step 4. Solve an algebraic equation containing radicals

If you need to solve these types of equations, then you will need to find a way to squared both sides of the equation so that the radicals disappear and then solve your equation normally. Here is an example:

  • √ (2x + 9) - 5 = 0

    First, move anything that is not under the radical into the other side of the equation

  • √ (2x + 9) = 5
  • Then raise each member squared in order to remove the radicals:
  • (√ (2x + 9))2 = 52 =
  • 2x + 9 = 25

    Now solve the equation as a normal equation, grouping the constants together and isolating the variable

  • 2x = 25 - 9 =
  • 2x = 16
  • x = 8
Solve an Algebraic Expression Step 10

Step 5. Solve an algebraic equation containing absolute values

The absolute value of a number represents its numerical value without taking into account its sign (positive or negative); the absolute value is therefore always positive. For example, the absolute value of -3 (which we write | -3 |), is simply 3. To solve this type of equation, we must first pass the absolute value to one side of the equation then solve the equation twice, once when what's in the absolute value is positive and once when what's in the absolute value is negative. Here is how to do it.

  • Here's how to solve an equation containing an absolute value by first isolating the absolute value then removing it and solving as we explained:

    | 4x +2 | - 6 = 8 =

  • | 4x +2 | = 8 + 6 =
  • | 4x +2 | = 14 =
  • 4x + 2 = 14 =
  • 4x = 12
  • x = 3

    Now solve again by reversing the signs of the terms on the other side of the equation (the side where the absolute value was not at the beginning):

  • | 4x +2 | = 14 =
  • 4x + 2 = -14
  • 4x = -14 -2
  • 4x = -16
  • 4x / 4 = -16/4 =
  • x = -4

    Now write the solutions on your piece of paper: x = -4 or 3

Advice

  • To check your results, go to wolfram-alpha.com. The site gives you the solution (or them) and also optionally allows you to see the stages of the calculation.
  • When you're done, replace the variable with the value you found and verify that both members of the equation are the same. If so, congratulations! You have just solved an algebraic equation.

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