Algebra Basics - Linear Equations
Introduction
In this article we are going to look at algebra basics and what this involves. Generally we will look at:
- Solving simple linear equations where the variable appears on only one side
- Solving equations by doing the same on both sides
- Solving equations with variables on both sides
Algebra Basics - Getting Started
Suppose that you are asked to think of a number, let us say x. You now double this number, so you have 2x. Suppose you now add 3, so you have the expression 2x+3.
Let us now say that the expression above is equal to 11. Then we then have the equation 2x+3=11. This is an equation that we can solve for x.
So how can we solve the equation 2x+3=11? Well one of the rules of algebra states that what we do to one side of the equation we must do to the other side. This is to keep everything balanced just like a see-saw.
Well what we need to do is the opposite of what was done in the introduction. You will recall that there are two things that happened:
1st: A number was doubled (multiplied by two)
2nd: An addition of 3 was performed
So to solve our equation we must undo what has been done:
2x+3=11
Subtract 3 from both sides
Divide by 2 on both sides to give:
And this is the final answer.
Algebra Basics - How to know if you are right?
How can we be sure that the answer of x=4 is actually correct?
Well one of the great things about equations is that you can actually check your solution by substituting it into the equation itself.
Our equation is 2x+3 and if we replace x with 4 we will have:
2 x 4+3= 8+3=11, which is the correct answer!
- Example
Try solving this question by covering up the solution and then comparing your answer.
3b+7=22
Solution
Subtract 7 from both sides
Divide by 3 on both sides to give:
Check: 3 x 5 + 7 = 15 + 7 = 22, which gives the correct answer!
- Dealing with negative numbers
Sometimes you might have to deal with negative numbers. Remember that if a negative number is multiplied or divided by another negative number the answer is positive. But if a negative number is multiplied or divided by another positive number then the answer is still negative.
The following examples will help to clarify this.
Subtract 8 from both sides
Dividing by 2 on both sides to give:
Check: 2 x (-3) + 8 = -6 + 8 = 2, which gives the correct answer!
- Solving equations with fractions
From what we have seen so far, when it comes to solving equations you need to “undo” it in order to find the solution. The same also applies to equations that contain fractions as we will see in the following example.
Add 7 to both sides of the equation
Multiply both sides by 3 to give:
Check: 903– 7=30-7=21, which is the correct answer !
- How to solve equations when variables appear on both sides
Here we “collect like terms”. We want variables on one side and constants on another.
Example: Solve the following equation 6x+4=3x-2
Method 1
Subtract 3x from both sides
Subtract 4 from both sides
Divide by 3 on both sides to give:
Method 2
What if we subtract 6x from both sides?
Adding 2 to both sides:
Dividing by -3 on both sides:
Understanding algebra basics is an important part of GCSE Maths especially if you are aiming for the higher paper and looking for a grade of no less than a 7. Should you be thinking of pursuing maths into A Level then a solid grasp of algebra is essential as there are so many algebraic skills that you will be using throughout your course.
During a GCSE Maths Easter Revision course which takes place over 3 days, algebra will be a topic that will be covered ranging from the basics, equations, forming equations, simultaneous equations as well as quadratics.
There will also be the opportunity for you to understand how to maximise your marks as there will be an assessment for one hour that looks at your working, layout etc.
Algebra is often considered a horrible topic but the rules of algebra simply follow the rules of arithmetic. Quite often the best strategy is to put things into context such as “1 cat + 1 cat + 1 cat = 3 cats” and there is no difference to this than doing “c + c + c = 3c”.
What is important to remember is that with an area of collecting like terms you can only add and subtract terms that are the same.
Keep looking at our articles that go over topics such as algebra and in no time you will be an expert in GCSE Maths!
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