Lesson video

Hi, I'm Mrs. Dennett, and in today's lesson, we're going to be solving linear simultaneous equations where you need to multiply both of the equations.

Before we start this question, let us remind ourselves about coefficients.

The coefficients of P are three in equation A, and four in equation B.

The coefficients of Q are two and five.

We want to make either the P coefficients the same, or the Q coefficients the same.

Firstly, we're going to label our equations, so we know which one we're talking about.

Let's start with P.

We need to make the coefficient the same by multiplying each equation.

Think about multiples of three and four.

We can use any common multiple, but it is much easier to choose the lowest common multiple, to keep the size of the number to a minimum for our calculations.

So we multiply everything in equation A by four, to get 12 P, and every term in equation B by three, to get 12 P.

The coefficients of P are both 12, which is what we wanted.

We now have the new equations, so we label them C and D, and we have answered the question.

Alternatively, we could have chosen to make the Q coefficients the same.

Consider multiples of two and five.

10 is the lowest common multiple.

So multiply equation A by five, and equation B by two, this produces two new equations, as you can see.

This time, the Q coefficients are both 10, which would also be an acceptable answer.

It doesn't matter that one term is positive, and that the other term is negative, only that the coefficients are the same.

Here is a question for you to try.

Pause the video to complete the task and restart once you are finished.

Here are the answers.

You have two options here.

Make the coefficients of P the same, multiplying by two, and then by five, or make the coefficients of Q the same multiplying the first equation by five and the second equation by four.

Now we're going to look at solving a pair of simultaneous equations.

Let's first remind ourselves of the coefficients.

The coefficients of P are three and four, and the coefficients of Q are two and five.

We label the equations A and B, so we know which one we're referring to.

I have decided to make the P coefficients the same.

So I multiply the first equation by four, and the second equation by three.

Here are my two new equations.

I've labelled them C and D, so I know which one I'm referring to.

We now need to eliminate the P's to find out what Q is.

We can use addition or subtraction to do this.

Here, the P terms are both positive.

The signs are the same, so we subtract, C take away D.

12 P minus 12 P is zero.

We've eliminated the P's, that is what we wanted.

Then we do 8 Q take away negative 15 Q, which is 23 Q.

Remember, subtracting a negative is just like adding.

Then, we do 44 take away 159, which is negative 115.

Be very careful about the order in which you subtract.

Remember we're doing equation C takeaway equation D.

Be very aware of those sneaky negative numbers, especially when subtracting.

Now we can solve to find Q.

We do minus 115 divided by 23, which is negative five.

Q is negative five.

And now that we know the value of Q, we can substitute this entire equation A or equation B, whichever you think will be easiest.

I've chosen to substitute into equation A because it looks like there'll be smaller numbers and they'll be positive too.

We can solve, taking care with the negative five, to find that P is equal to seven.

Always check that your solutions are correct using the other equation.

So we substitute P equals seven and Q equals negative five into equation B.

Let's see if it works.

This does work out correctly, so now all I need to do is state at my answer clearly, and I'm done.

This is not the only method you could have used.

We could have made the Q coefficients the same, and then this would have involved elimination by addition, and may have been a little easier to do in terms of calculations, because of all those horrible negative numbers.

Either way we would still have got the solution P equals seven and Q equals negative five.

To summarise, there are seven simple steps to follow to help you successfully solve simultaneous equations in this way.

Firstly, remember to label each equation.

Then, make a pair of coefficients equal using multiplication.

Eliminate that pair of coefficients by addition or subtraction.

Remember same sign subtract.

That can be really useful in helping you to decide whether to add or subtract the coefficients.

Solve to find one solution.

Then substitute this solution into one of the original equations.

Solve to find the second solution.

And finally, check that both solutions work in the other equation.

This is absolutely crucial and can be really, really useful in helping you to spot errors.

Then give yourself a huge pat on the back for successfully completing all of these steps.

Here's a question for you to try.

Pause the video to complete the task and restart when you are finished.

Here are the answers.

How did you get on with your first attempt at following all seven steps? The great thing about these questions is that there are a couple of methods for you to choose from.

You can choose which coefficients you want to make the same.

I always like to keep the numbers smaller, where possible, but this isn't essential.

Think about how many skills you are using in this question.

All four operations, add, subtract, multiply, divide, substitution, and then solving as well.

There really is so much going on here.

So let's practise a few more.

Here's a few more questions for you to practise.

Follow each step carefully and be aware of negative numbers.

Watch out for a non-integer solution to question B.

Pause the video now to complete these questions and restart when you are finished.

Here are the solutions.

Part C was a little trickier because you needed to expand the bracket in the second equation or divide by two throughout to get four X minus Y equals three before you solved it.

Here is a final question for you to try.

You will need to form two equations before you solve.

Pause the video to complete the task and restart when you are finished, Here are the two numbers.

You can use simultaneous equations to answer this question, using, for example, two letters, X and Y, to stand for the numbers.

X plus Y is 13.

The second equation is four X plus three Y equals 45.

because when I multiply a number by four, and the other one by three, I get 45.

I would have to multiply X plus Y equals 13 by three.

This gives me three X plus three, Y equals 39.

This means that the Y coefficients will be the same.

They both have the same sign, which is positive three Y, and so I would have to subtract the equations to find that X equals six.

Subtract this from 13 to find Y, which is seven.

So the two numbers are six and seven.

That's all for this lesson.

Remember to take the exit quiz.

Thank you for watching.