Solving Linear Equations
Solving Linear Equations – Introduction
What we are going to look at here is how to solving linear equations and very simple equations. This will include equations with just unknown, unknowns that appear on both sides, equations that have brackets and also equations that have fractions.
The questions and methods that are discussed are suitable for all exam boards such as Edexcel, AQA and OCR. You should try the questions that we have posted and when looking at the solution to see if you are right, you should also look at the comments which highlight the typical common mistakes that are made.
What are linear equations?
Linear equations are equations that contain just one unknown which is also known as a variable and these variables are all to the power one.
They also all have equal signs and this means that everything on the left is equal to what is on the right. A typical example of a linear equation is the equation of a straight line which you will know is of the form y=m x+c. You can also see linear equations written in the form a x+b y+c=0.
Solving Linear equations - Examples
Q1.
Solve 3 x=18
You can often think of these questions in terms of “you think of a number, you multiply it by 3 and the answer is 18. What number did you think of?”
Looking at the worded question, you will hopefully recognise that to solve this question you work backwards and so do the opposite process.
In other words you would do 18 \div 3=6.
And this is generally the process when it comes to solving linear equations, to undo it, you have to do the opposite of what you see.
Q2.
Solve x+2 x=12
To do this question you first need to collect like terms. This will then give us:
\begin{aligned} & 3 x=12 \\ & x=4 \end{aligned}A common mistake here would be that of arithmetic such as dividing 12 by 3.
Q3.
Solve 2 y-1=13
Remember you want to do the opposite of what you see which in this case is going to mean adding 1 and dividing by 2. This will give the following:
\begin{aligned} & 2 y=14 \\ & y=7 \end{aligned}Another method that some students may use is that of trial and substitution. This can lead to errors and confusion and for such a simple question, such a technique should be avoided.
More Solving Linear Equations
Q4.
Here we have a worded question and it would be good to be able to rewrite the above in terms of a linear equation. So let the number that is thought be \chi so:
3 x+19=61Of course you do not need to do this but you can see that the question gives a linear equation, there are less words to consider, and you can follow the process of doing the opposite as seen with the previous examples so far.
\begin{aligned} & 3 x=42 \\ & x=14 \end{aligned}Again, another approach would be to use trial and improvement but this can lead to errors.
Q5.
Simplify 4 \times p \times q
There is no equation to solve here, you are just being asked to simplify the expression that is given to you.
The answer here is simply 4pq. When doing multiplication of variables, the multiplication symbol is generally not written, for the purpose of neatness.
Not simplifying the expression would mean writing something such as 4 p \times q or 4 \times p q.
These types of answers would be incorrect.
Q6.
a) Solve 4 x=16
For this question you do the opposite which is division to give a final answer of x = 4. The main thing to be careful of is arithmetic.
b) Solve c-8=11
Remember to do the opposite, so you need to +8, to give c = 19
c) Solve \frac{y}{5}=2
The left side shows a fraction which is the same as division. Think of the question as “You think of a number you divide it by 5 and the answer is 2. What number did you think of?”
To do the opposite you would multiply the 2 and the 5 together to give y = 10.
These are the main types of questions that you can expect to appear on a foundation GCSE maths paper and the skills are still needed if you are sitting the GCSE Higher Paper. We will create another post very soon that will look at dealing with linear equations that show variables on both sides as well as linear equations that have brackets.