Lesson video

Hi, am Mrs Dennet and in today's lesson, we're going to be looking at linear simultaneous equations, where you have to multiply one of the equations in order to solve it.

We're going to start this lesson by thinking about these two equations.

What is the same and what is different? Pause the video to complete the task and restart when you finished.

Here are the answers.

Did you notice that the second equation is double the first equation.

When you multiply or divide each term in an equation are the same number, the values of the letters always stay the same.

You can prove this to yourself by substituting in a value for f and a value for g into the equations.

For example, if x equals five g equals 15 The equations have changed, but not the values of f and g.

Five plus 15 is 20 , 10 plus 30 is 40.

In order to solve a pair of linear equations, we need to use what we have just looked at, in the previous question.

If we multiply each term in an equation by the same number, the values of x and y will not change.

Here is equation A going to multiply it by two, we get six x plus 14 y equals 10.

We'll call that equation B.

Here are a couple more examples for you.

Multiply an equation A by five gives us this equation and multiply an equation A by naught point five gives us this equation.

When we multiply the x and y terms by a number, we are changing the coefficient In this case, the number in front of x, of a number in front of y.

Notice that when we're multiplying an equation, it is useful to label it with a letter and a reminder of what you multiply by.

Here are a pair of equations.

Let's label them.

The next thing we must do in order to solve is make a pair of x coefficients or a pair of y coefficients the same.

Which pair of coefficients will it be easier to equate? Definitely the x coefficients, we just need to multiply equation B by four, we get four x plus 12, y equals eight and the coefficients of x and are the same.

Here's another example.

Which equation should we multiply this time and by what number? This is better to meet the Y coefficients the same, multiply equation A by two.

Now, both of the Y coefficients are 10.

Here are some questions for you to try.

One tip I would give you, is to make sure that the you multiply every term in the equation by your chosen number.

Otherwise, you will change the solution to the equations.

And we don't want to do that.

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

Here are the answers.

For the first pair of equations, we can multiply equation A by two to make the x coefficients the same, or B by two to make the Y coefficients the same As you can see, sometimes you have a choice as to which coefficients you want to change.

Just make sure that you multiply each term by your chosen value.

We are going to practise one more scale before we start to solve simultaneous equations.

I would like to give you a big hint for this question.

Use substitution.

Pause the video to have a go at this task and restart when you are finished.

Here is the answer.

f equals 10.

And g equals nine is not a solution to these equations.

10 plus nine equals 19.

But 30 takeaway 18 is not equal to eight.

This method of substitution will be really useful later on, when you require to check your answers after you've solved a pair of equations.

Now it's time to start solving simultaneous equations.

We'll start by using bar models to help us.

Firstly, label each equation.

Here are the bar models to represent each equation, we need to make a pair of coefficients the same.

This will give us either the same amount of r boxes, or the same amount of s boxes in our bar model.

Which equation should we multiply by how much? It is simplest to multiply equation A by two.

This will give us eight r plus two s equals 46.

The s coefficients are the same.

We can now eliminate to s.

We use addition or subtraction to do this.

Here, the signs are the same, both positive so we have to use subtraction to eliminate the s turns.

You can do this in any order, but to avoid negatives, I'm going to subtract equation B from equation C So we get eight r minus three r, leaving us with five r.

Two s take away two s, which leaves us with zero s, and 46 take away 31, which leaves us with 50.

This simplifies to five r equals 50 r must be three, we can substitute r equals three into equation A to find s four times three gives us 12 plus s equals 23.

So s must be 11.

We can check the solutions by substituting r equals three and S equals 11 into equation B.

Three times three or two times 11 equals 31 nine and 22 gives us 31.

So we are correct.

Here is a question for you to try.

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

Here are the answers.

first equation has been doubled to give to r plus two s equals 40.

The coefficients are the same, both positive.

So we subtract one equation from the other.

Three r minus two r gives us r.

And the s turns are eliminated 44 takeaway 40 leaves us with four, so r is equal to four.

Here are some more questions for you to try.

You can use bar models or just algebra calculations to help you answer them.

Pause the video and restart when you finished Here are the answers.

You will no doubt have used a variety of methods to solve these pairs of equations.

For each one, you may have made the x coefficient the same, or the Y coefficients the same and this will then determined whether you added or subtracted the equations.

Notice we can have a decimal fractional or even negative solutions as well.

This question requires you to form two equations, before you can solve them.

Good luck.

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

Here's the answer.

We formed two equations using the information given we can then make the c coefficients the same.

Multiply an equation one by two, or the s coefficients the same.

Multiply an equation one by four.

It's up to you which method you choose.

Either way, we then subtract the first equation from the second and solve.

We can see there are four sheep in the field.

That's all for today's lesson, remember to the exit quiz.

Thank you for watching.