﻿Hey everyone, this is Ms. Bridgett and in today's lesson, we are gunna be looking at whether we should solve simultaneous equations using elimination, or whether we should be solving them using substitution. You are gunna need a pen, you are gunna need some paper, make sure that you have removed any distractions. If you are ready, let's get started. Here, we've got two simultaneous equations, 2x subtract y equals 7 and x is equal to 2 plus 2y. Carla used substitution to solve these simultaneous equations. Zackie solved them using elimination. Have a go at solving them, using both of these methods. After you have done this, stop and think about which one you'd prefer for this set, this pair of equations. Watch the video, and have a go! OK, let's have a look at Carla's method. So she used substitution. This might not have been exactly the same way that you've done it, but it is still worth us taking a look at it. So, if we look at our first equation 2x subtract y equals to 7, we can use equation two, and substitute equation two into it. That leaves us with an equation with just one unknown in it, y. From there, we can simplify, and we can solve. And we get to y is equal to 1. Once you have got y is equal to 1, we can go back to one of the equations, so Carla went back to equation two, and we can find the value of x. Let's have a look at Zackie's method. So, he used elimination. So what he did, was he took the second equation, and he rearranged it. So, the second equation says x is equal to 2 plus 2y. He rearranged that so that both of the unknowns were on the left-hand side of the equation. X subtract 2y is equal to 2. We are going to call that equation three, to make it easier for us to refer to. Let's get back to equation one. We've got 2x subtract y is equal to 7. Now, when I compare those two equations at the moment, you can see that neither the x's nor the y's have the same absolute value for the coefficient. So, what Zackie then did was took equation three, and doubled it. So, when we double equation three, we get 2x subtract 4y is equal to 4. I am going to call that equation four. Now, if we just look at equation one, and equation four, you can see that the coefficient of x is the same. So, that means that we can subtract these equations to eliminate the x's. So if we subtract them, we end up with 3y is equal to 3, which means that we know that y is equal to 1. And we can go back to one of our equations, to work out the value of x. Now, either way, we get exactly the same solution. Now I am wondering which one of your methods you prefered. As I said, you might have solved them slightly differently to the way that Carla and Zackie did, but I have to say, for this particular instance, I think that substitution was the most efficient method. In the last example, the pair of equations that we had, it was more efficient to solve them using substitution rather than elimination. This isn't always the case. What I would like you to do it to just take a look at these two pairs of simultaneous equations, and decide what you think the most efficient method would be, and why. Pause the video, and have a think. OK, so I think that the equation on the left simultaneous equations on the left, are easier to solve using substitution. And the reason for that is that we can, we can see that one of the equations has got x as the subject. So, it is easy for us to take that equation, and substitute it into the top one. And the equation on the right, neither p nor q is the subject of the equation. So if we wanted to use substitution, we would have to do some rearranging. And they have been nicely lined up for us, and we can also see that the coefficients appear the same. Because the coefficients appear exactly the same, we can just subtract those equations, and we can eliminate the p fairly easily. Sometimes it is easier for us to solve a pair of simultaneous equations by using substitution, and sometimes it's easier for us to solve a pair of simultaneous equations using elimination. What I'd like you to do in this next task, is to take these simultaneous equations, and sort them into two groups. Those that you prefer to solve using substitution, and those that you prefer to solve using elimination, be prepared to justify your answers, but you don't need to solve them. Pause the video, and off you go. OK, now you may disagree with me on this, but looking at that first set of equations, I think that the best way to solve this would be by elimination. I've got x's and y's, I've got both of those unknowns on one side of the equation, and if you look at the x's, the coefficient of them is exactly the same. So, I think that we could subtract these equations. So, if we subtract these equations, the x's would eliminate, and would end up with 6y is equal to 4. So, the 2y subtract -4 would give me the 6y. If we move onto the second set of equations, that very top equation, is in the form y equals. So, because it is in the form y equals, it's easy for us to substitute in, we don't have to any rearranging before we substitute into the second equation. So, I would use substitution here. And, that will give me, from the second equation, the following, and that is an equation with just one unknown in it. Let's have a look at the third pair of simultaneous equations. Now, I don't what you thought about this one, but to begin with, I thought about substitution. So, I rearranged the top equation, so that x was the subject. So, rather than being x plus y is equal to 3, it's x equals 3 subtract y. I then substituted that into the second equation, and got an equation with just one unknown in it. Then I thought, maybe elimination wouldn't have been so bad. So, for elimination, I kept the top equation as it was, and I rearranged the bottom equation. And when I did that, you can see that the absolute value of the coefficients of the y's is 1. So, we can add to those equations, and it will eliminate the y's. Now, I'm not sure which one of those is more difficult than the other, so with this equation, sets of equations, I don't think it matters which one that we do. For the final set of equations, I also maybe think that this one is a bit little bit controversial as well, so to begin with, I thought definitely substitution, definitely substitution, and I took the second equation, and I substituted it into the first. But then I stopped to look again, and I thought, actually, I could subtract them. If I subtract the bottom equation from the top equation, I end up with 0 is equal to 5 minus 2x, I end up with an equation with just one unknown in it. So, maybe elimination is slightly easier here, but I am not sure there is an awful lot in it. Your final task, I would like you to create three pairs of simultaneous equations of your own. Use the values of p and q that you can see on the left. So p is equal to 3, and q is equal to 7. Now, when you are writing your pairs of simultaneous equations, I'd like you to make one set that is easier to solve by elimination, one set that is easier to solve by substitution, and one set that is impossible to solve. If you remember a couple of lessons ago, we looked at systems of equations where it wasn't possible to find a solution. Pause the video, and off you go. OK, here with this set of equations that I came up with, so, first of all, I created a set of equations that were going to be easier to solve by elimination. So, I wrote two equations, where the unknowns were both on one side of the equation, and also where the absolute value of the coefficient of one of the unknowns was the same. So here, I've got 2p plus q is equal to 13, and p subtract q is equal to -4. So you can see that if I were to add those, the q's would be eliminated. My next pair of equations, I wanted to make it easier to solve by substitution, so here they are. P is equal to q subtract 4, and p plus 5q is equal to 38. So, the way that I created these was to make p the subject of one of them. My final set of equations, that were impossible to solve, are these two. So I created an equation involving p and q, so, q is equal to p plus 4, 7 is equal to 3 plus 4, and then I multiply it by 10. Remember that because they are multiples of one another, it's not containing any new information. This is the same information repeated, and therefore, we haven't got enough information for us to be able to identify what those unknowns are. So that's the end of today's lesson, thank you as always for all of your time, and in the next lesson we will be carrying on with simultaneous equations.

In this lesson, we will compare the different ways to solve simultaneous equations. We will determine when and where to use different strategies, and develop a sense of the advantages and disadvantages of each method.

Maths

Y9 Maths

Key Stage 3

Year 9

Substitution or Elimination

Solving linear simultaneous equations algebraically

Algebra

THEME

UNIT

Reviewing solving equations

Equivalent equations

Linear relationships

Combining algebraic relationships

Exploring expressions with two variables

Exploring systems of equations

Solving simultaneous equations

Writing simultaneous equations algebraically

Checking the solution of simultaneous equations

Solving a simultaneous equation algebraically

Eliminating a variable

Manipulating systems of equations

Solving word problems

Using substitution to solve simultaneous equations

Simultaneous equations by trial and error

Let p be the number of protractors bought. If each protractor costs £2, which of the following represents the total amount spent on protractors (in pounds)?

If x=2 and y=3, what is the value of 2x+3y?

Use substitution to decide which of the following is the solution to x+y=3 and 5y+3x=15.

What number am I thinking of, if half of the sum of my number and 4 is the same as 3 lots of my number subtracted by 2?

Fill in the gaps: The "n" in the nth term represents the ____.

Which sequence is formed as a result of adding "4, 9, 14, 19, ..." and "12, 8, 4, 0, ..." together?

I want to buy 3 rulers and 3 protractors. If they cost 40 pence each, how much will I spend?

If p+1=6 and p+q+1=7, what is the value of q?

4x +8y=116 AND 4x +4y=68. Solve

Is there a solution with x value between 0 and 1?

Look at the picture below and then fill in the gap: If two equal expressions are both __________ by the same quantity, then the two resulting expressions will be equal.

3x+2y=5 AND 3x-2y=7.  If I add these two equations together, which unknown will be eliminated?

Lesson overview:Substitution or Elimination

Core Content

Solving linear simultaneous equations algebraically: