Lesson 7: More Equations. Word Problems (II)

February 1, 2011

4.2. Solving equations with brackets

When equations involve brackets then we have to:

1º.  Eliminate brackets

2º.  Group terms with  in the left-hand side and numbers in the right-hand side.

3º.  Collect like terms

4º.  Isolate  in the left-hand side, that results in  equals one number. This number is the solution.

5º.  Check the solution.

NOTE

Remember that if a – sign precede brackets when you eliminate brackets, all the terms inside changes their signs. For instance;  –2(x – 3) = –2x + 3

View the following video on solving equations with brackets

4.3. Solving equations with denominators

When equations have denominators you have to:

1º.  Eliminate denominator by multiplying both sides by the HCF of all the denominators that appear

2º.  Eliminate brackets

3º.  Group terms with  in the left-hand side and numbers in the right-hand side.

4º.  Collect like terms

5º.  Isolate  in the left-hand side, that results in  equals one number. This number is the solution.

6º. Check the solution.

Practise on this web pages equations with brackets and denominators, easy, I believe

In this other link you can find harder equations. Solve only linear equations. The others have radicals and algebraic fractions. When you click on solution you will find the example solved. If you go at the bottom of the page where the solution is you will find other example (click on it). If you would like to work with some similar problems click on problem.

Finally a video. Try to understand. If you are in a hurry write to me.

4.4. Equations that are identities

In some exercises, you have to solve an equation that results to be an identity. Let’s see the following example.

Example

Solve 2x – 2 =2(x +2)  – 6

First we eliminate brackets: 2x – 2 =2x +4  – 6

Transposing terms: 2x – 2x = 4  – 6 + 2

Collecting like terms: (2 – 2)x = 0 →0·x = 0

But any number (known or unknown) multiplied by zero gives zero, so:

We deduce that the equality is always true. This equation is true, regardless of the value of which it indicates it is an identity.

We can say that an equation of the form , is an identity.

4.5. Equations without solutions

Some equations have no solutions. They are named inconsistent equations. Here it is an example.

Example

Solve x – 3 = 2 + x

Transposing terms: x – x = 2 + 3

Collecting like terms (1 – 1)·x = 5  → 0·x = 0

But any number (known or unknown) multiplied by zero gives zero, so: 0 = 5

In the example you found that . What does this mean? Obviously, this equations cannot be true, regardless of the value of which it would indicate it is an identity. The equality is always false.

We can say that an equation of  the form   0·x = number, doesn’t have a solution.

5. Solving words problems with linear equations

Most of the time when someone says “word problems” there is automatic panic. By setting up you can be successful with word problems. So what should you do? Here are some recommended steps:

1.  Read the problem carefully and understand what it is asking you to find. Usually, but not always, you can find this information at the end of the problem.
2. Assign a variable to the quantity you are trying to find. Most people choose to use x, but feel free to use any variable you like. For example, if you are being asked to find a number, some students like to use the variable n. It is your choice.
3. Write down what the variable represents. At the time you decide what the variable will represent, you may think there is no need to write that down in words. However, by the time you read the problem several more times and solve the equation, it is easy to forget where you started.
4. Re-read the problem and write an equation for the quantities given in the problem. Make yourself the following questions: What do we know? (DATA) How can I translate the information into algebra?
5. Solve the equation.
6. Answer the question in the problem. Just because you found an answer to your equation does not necessarily mean you are finished with the problem. Many times you will need to take the answer you get from the equation and use it in some other way to answer the question originally given in the problem.
7. Check your solution. Your answer should not only make sense logically, but it should also make the equation true. If you are asked for a time value and end up with a negative number, this should indicate that you have made an error somewhere. If you substitute these unreasonable answers into the equation you used in step 4 and it makes the equation true, then you should re-think the validity of your equation.

The links below contain different types of problems. Try the type you want to practise.

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