[Please watch the video attached at the end of this blog for a visual explanation of this topic]
What are “Simultaneous Equations”?
Two or more algebraic equations that have similar variables, such as x and y, a and b, etc. They are known as simultaneous equations because both equations are solved at the same time, i.e. simultaneously.
These variables x and y are usually unknown terms and by solving the said equations, we can find the answers once these two or more variables are linked together. Therefore, by using the equations, you can then solve the two letters/variables given. That is the purpose of simultaneous equations.
In order to solve said simultaneous equations, there are two renowned methods we use.
- Substitution Method
- Elimination Method
- Both these methods are crucial for exams.
The substitution method is generally used when you have expressions where it is easy to rearrange the given expression in the form of y = ? or x = ?
3x + y = 19 and x + y = 9
For example, the second equation above can be rearranged into:
y = 9 – x
After this step is done, this second equation can be substituted into the first equation and it can be solved to find x.
In the above example, we can see that the second expression looks simpler and easier to rearrange. Therefore, we can rearrange it to y = 9 – x. (This has been done by following simple algebraic rules that can be revised here!)
Then we consider the first equation (3x + y = 19), and instead of the y, we replace it with the expression that has been rearranged ( y = 9 – x). By doing that, we can solve and find the value for x, because now y has been removed from the equation.It is now a simple linear equation.
3x + y = 19
As y = 9 – x, then 3x + y = 19 becomes 3x + 9 – x = 19.
This can be simplified into
2x + 9 = 19
2x = 19 – 9 = 10
2x = 10 (both sides need to be divided by 2)
x = 10/2 = 5
This is NOT the end of your work however. Since you have found the value of x, you must now use it to find the value of y.
We can consider the second equation we rearranged (y = 9 – x), and here, instead of x, we are going to substitute it with the value of x that we found earlier.
y = 9 – x
As x = 5, y = 9 – 5 = 4.
This way, we have found what both x and y stand for. In any simultaneous equation, you must provide an answer for both the unknown variables, both x and y, or a and b, or any two letters which are involved.
- This is a very common mistake that most students make. They think that their job is done after finding x or y. However, in order to obtain full marks at the exam, you must solve both variables.
In elimination, the purpose is to cut off one of the variables so we can simplify and solve it.
In order to do this, we must figure out which factor/ number we must multiply an equation or perhaps both equations by. This is done so that the number in front of x or the number in front of y is the same.
2x – y = 7 and 3x + 2y = 7
In the given example, we can take the first equation and multiply the entire equation by 2.
This will result in 4x – 2y = 14. Third is the 3rd equation.
- Remember, when you are multiplying an equation with a constant, all terms of the equation as well as both sides of the equation must be multiplied in order for it to be mathematically correct.
The next step is to eliminate one term (in this case, it would be easier to eliminate y)
When we consider the 2nd and 3rd equations, since they have + 2y and – 2y in them respectively, we can cancel those two terms by adding the two equations together.
2nd equation → 3x + 2y = 7
3rd equation → 4x – 2y = 14
2nd equation + 3rd equation:
3x + 2y + 4x – 2y = 7 + 14
7x = 21
x = 3
Once again, we must take the value of x and substitute it in one of the equations given to find the value of y.
If we consider the 1st equation to make things easier:
2x – y = 7
Since x = 3
(2 ✖ 3) – y = 7
y = 6 – 7 = – 1
Important things to remember when attempting Simultaneous Equations questions
- Remember by which constant you have multiplied the equation with, and double check to see if all the terms and both sides of the equation have been multiplied by the said constant. This is done to cancel two terms off.
- If the signs are different, then we add the two equations, but if the signs are the same, then we subtract the two equations. (in this case, one of them was positive 2y (+ 2y) and the other was negative 2y (– 2y) that is the reason we added the two equations)
Master Simultaneous Equations!
Simultaneous Equations are possibly the most fun lesson in Algebra. The only thing you must do is pay close attention to what you are writing, ensuring that you do not mix up the positive and negative signs and solving both the variables.
However, like all lessons, you can become great at this particular area in Mathematics if you practise as many questions as you can. Some questions can be found here as well, and you can time your answers to see if you can stick to the time limit given.
If you are struggling with IGCSE revision or the Mathematics subject in particular, you can reach out to us at Tutopiya to join revision sessions or find yourself the right tutor for you.
Watch the video below for a visual explanation of the lesson on mastering simultaneous equations. Don’t forget that there is a timed quiz to test whether you have understood the lesson!