Lesson video

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Hi, I'm Mrs. Dennett, and in today's lesson, we're going to be solving simultaneous equations where you first have to rearrange.

Before you start this lesson, you should be fairly confident at solving simultaneous equations.

You can always revisit this skill before you continue.

We're going to start by recapping how to rearrange equations.

So far, you will have seen simultaneous equations that look like this.

The letters, in this example x and y, are nicely aligned with the number after the equal sign.

Sometimes, however, you may get a set of equations that looks like this.

The x's are aligned but the y's and the number terms are not.

This will make it difficult to add or subtract terms when we are eliminating one of the unknowns, in this case x or y.

So we have to rearrange one or both of the equations before we start to solve.

Firstly, let's label the equations a and b.

The x's are already aligned.

In fact, equation b has x's then y's equal to 17.

So we don't need to rearrange it.

We just need to think about the plus five and y, which is also a positive term.

We use inverse operations to help us to rearrange.

We also have to remember to follow the rules of order of operations.

We start by subtracting y and then we can take away five.

We could have done this in any order as we were just dealing with addition and subtraction.

So we could have taken away five first from both sides of the equation, and then take away y.

We would have got the same answer.

Now, equation a looks like equation b.

X, y, and the number terms are aligned and the equations are ready to be solved.

There maybe occasions where you have to rearrange both equations, like this example here.

And you may encounter some fractions too.

It's nearly always best to eliminate the fraction part first.

So here you would multiply by four to get three y minus three equals four x, and then continue to rearrange.

Here's a question for you to try.

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

Sara has made two common errors.

In the first equation, she needed to subtract seven, leaving negative seven on the right hand side.

In the second equation, Sara has correctly tried to multiply both sides of the equation by seven.

The right hand side is correct, but the left hand side should have eliminated the fraction.

By multiply the numerator, she has in fact multiplied the equation by seven times seven.

The numerator should remain as five y take away three.

Now we're going to recap how to solve simultaneous equations.

But be careful, what do we need to do first? Let's level our equations and then rearrange them so that we can eliminate one of the unknowns.

I add three p to both sides of the equations for a, and for b, I need to take away five q, add 53, and I end up with my p's and q's aligned.

Do we have a pair of equal coefficients? No.

We need to make a pair of coefficients the same.

We can choose the coefficients of p or the coefficients of q.

I'm going to choose the coefficients of q.

I'm going to multiply by five and by two.

Here are my two new equations.

I've labelled these for clarity.

To eliminate q, we must now add.

The signs are different, positive 10 and negative 10.

So adding the two equations will give us zero q.

And that is what we want to achieve.

Add the p terms and the number terms, and we get 23p equals 161.

We can see that p is equal to seven.

We put this into equation a.

You could put it into equation b if you prefer.

And we solve to find q.

Q is minus five.

Let's perform a quick check using the original equations.

We could use c or d here, but I'd prefer to use a and b in case I made a mistake whilst rearranging.

I've chosen to substitute p equals seven and q equals negative five into equation b.

We can see that this does work.

We are correct.

So the solutions to the equations are p equals seven and q equals negative five.

Here's a question for you to try.

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

Here is the answer.

Rearrange the first equation before you begin to solve two x minus y equals negative seven.

The x coefficients are the same, so we can subtract the equations, finding that y is equal to three.

Put this value into one of the original equations to get x equals negative two.

This is a quick reminder of the steps involved before you complete some more practise.

You label each equation, and then we need to check if we have to rearrange them first.

You make a pair of coefficients equal, if we need to, and then we eliminate that pair of coefficients using addition or subtraction.

If the signs are different, we need to add to eliminate.

If the signs of the same, both negative or both positive, we subtract.

I like to remember same sign subtract.

It sounds like a snake.

And now we need to solve to find one solution.

Once we've got our solution, we can substitute this into one of the original equations.

Solve to find the second solution and check that both solutions work in the other equation.

If they do, give yourself a pat on the back.

You've successfully managed to solve simultaneous equations.

Here's a question for you to try.

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

Here are the answers.

Both of these equations contain fractions.

Rearrange the first equation in part a to get minus three y plus x equals three before solving it.

For part b, rearrange the first equation to get three x plus four y equals 17.

And the second equation to get seven x minus five y equals minus 32.

You may have found variations of these instead, which is absolutely fine as long as your final answer is x equals minus one and y equals five.

Here is a question for you to try.

Pause the video to complete your task and restart when you're finished.

Here is the answer.

Opposite sides of a rectangle are equivalent.

So we form two equations to reflect this.

In order to solve in part b, we have to rearrange the second equation to four a minus two b equals 28.

We make the coefficients the same by multiplying one or both of the equations.

Solve and substitute to get a equals eight and b equals two.

That's all for this lesson.

Thanks for watching.