Solve equations with algebra
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How to solve linear equations in one variable using algebra.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to solve linear equations in one variable using algebra.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Algematics software. Click here for instructions.

This topic shows you how you can use Algematics to help you solve linear equations. You must decide what steps to do to solve a linear equation, but Algematics makes sure that each step is mathematically correct.

Theory:

A linear equation in one variable has a single unknown quantity called a variable represented by a letter. Eg: ‘x’, where ‘x’ is always to the power of 1. This means there is no ‘ x² ’ or ‘ x³ ’ in the equation.

The process of finding out the variable value that makes the equation true is called ‘solving’ the equation.

An equation is a statement that two quantities are equivalent.

For example, this linear equation:  x + 1 = 4 means that when we add 1 to the unknown value, ‘x’, the answer is equal to 4.

To solve linear equations, you add, subtract, multiply and divide both sides of the equation by numbers and variables, so that you end up with a single variable on one side and a single number on the other side.

 As long as you always do the same thing to BOTH sides of the equation, and do the operations in the correct order, you will get to the solution.

For this example, we only need to subtract 1 from both sides of the equation in order to isolate 'x' and solve the equation:

x + 1 - 1 = 4 - 1

Now simplifying both sides we have:

x + 0  = 3

So:

x  = 3

With some practice you will easily recognise what operations are required to solve an equation.

Here are possible ways of solving a variety of linear equation types.

Example 1, Solve for ‘x’ :

x + 1  =   -3

1. Subtract 1 from both sides:

x + 1 - 1 =   -3 - 1

2. Simplify both sides:

x   =   -4

Example 2, Solve for ‘x’ :

-2x   =   12

1. Divide both sides by -2:

2. Simplify both sides:

x   =   -6

Example 3, Solve for ‘x’ :

1. Multiply both sides by 3:

2. Simplify both sides:

x   =   -6

Example 4, Solve for ‘x’ :

2x + 1  =  -17

1. Subtract 1 from both sides:

2x + 1 - 1  =  -17 - 1

2. Simplify both sides:

2x  =  -18

3. Divide both sides by 2:

4. Simplify both sides:

x   =   -9

Example 5, Solve for ‘x’ :

1. Multiply both sides by 9:

2. Simplify both sides:

3x   =   36

3. Divide both sides by 3:

4. Simplify both sides:

x   =   12

Example 6, Solve for ‘x’ :

  1. Multiply both sides by 3:

  2. Simplify both sides:

 x + 1   =   21

  3. Subtract 1 from both sides:

 x + 1 - 1 =   21 - 1

  4. Simplify both sides:

x   =   20

Example 7, Solve for ‘x’ :

7(x - 1)  =  21

1. Divide both sides by 7:

2. Simplify both sides:

x - 1   =   3

3. Add 1 to both sides:

x - 1 + 1 =   3 + 1

4. Simplify both sides:

x   =   4

Example 8, Solve for ‘x’ :

1. Multiply both sides by 5:

2. Simplify both sides:

3(x - 1)   =   30

3. Divide both sides by 3:

4. Simplify both sides:

x - 1   =   10

5. Add 1 to both sides:

x - 1 + 1 =   10 + 1

6. Simplify both sides:

x   =   11

Example 9, Solve for ‘x’ :

5x + 2  =  2x + 17

1. Subtract 2x from both sides:

5x + 2 - 2x =   2x + 17 - 2x

2. Simplify both sides:

3x + 2   =   17

3. Subtract 2 from both sides:

3x + 2 - 2 =   17 - 2

4. Simplify both sides:

3x   =   15

5. Divide both sides by 3:

6. Simplify both sides:

x   =   5

Example 10, Solve for ‘x’ :

5(x - 4)  =  3x + 2

1. Expand brackets:

5x - 20   =   3x + 2

2. Subtract 3x from both sides:

5x - 20 - 3x =   3x + 2 - 3x

3. Simplify both sides:

2x - 20   =   2

4. Add 20 to both sides:

2x - 20 + 20 =   2 + 20

5. Simplify both sides:

2x   =   22

6. Divide both sides by 2:

7. Simplify both sides:

x   =   11

 

Download the free support file... We have created an Algematics document containing the completed examples from this topic. It also includes practice exercises to improve your skills.

File name:  'Solve equations with algebra.alg'   File size: 15kb
Click here
to download the file.

If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, click here for instructions.

 

Method:

IMPORTANT: This topic assumes that you know how to enter mathematical formulas into Algematics. Find out how by completing the three simple tutorials in the 'Getting Started' section of the Algematics program 'Help'.

First of all, work out a plan for solving the linear equation. You can use the examples above as a guide. Now you can use Algematics to help you find the solution.

Step 1  Enter the data

Click  and type the linear equation into the maths box in the data entry box.

If the ‘EMPTY’ message is not displayed between the blue buttons, click the  button until the message: ‘INSERT’ appears. If required, use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for divide.

 

       Maths...

   x + 1 = -3

 

 

 

and then click

Step 2  [Possibly] expand brackets

If you need to expand brackets, like in example 10 above, then click the  (expand) button.

Step 3  Perform +, -, ΄ and Έ operations

You should already have a strategy in mind for solving the equation, so it's just a matter of doing the correct operations in the correct order.

To do an operation, you:

a. Click on the input box and type the input quantity.

Each of these operators requires input.
For example, if you were going to add ‘2’ to both sides, you would type ‘2’ in the input box.

 

b. Click on the required operator button:

, , , ,

c. Click  (simplify all) to simplify the expression.

 

Repeat step 3 until you reach the solution.

If you make mistakes, press Ctrl+Delete on the keyboard to back-track.

 

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