In this video, we're going to cover how to solve simultaneous equations using a method called elimination. Simultaneous equations, remember, are pairs of equations, like these two, that you have to solve by finding the pair of x and y values that have solutions for both of them. So an x value and a y value, that would work for both of these equations.
Now, this method is kind of complicated when you first learn it, because it can involve a lot of steps. So in this video what we're going to do is go through this example here first, which is a relatively easy one and we'll hopefully get the idea across, and then we'll go through a more complicated one where we have to work through every step in detail. The basic idea with the elimination method is that we're going to try and combine these two equations into just one equation, but importantly when we do that we're going to try to eliminate either the x's or the y's at the same time.
so that we only have one unknown variable left in our equation. For example, because both of these equations have two y's, if we subtracted this whole second equation from the first one, then we'd eliminate the y's, because 2y minus 2y equals nothing, and then we'd only have the x terms and the numbers left, so we'd be able to solve it. If we give this a go, the first thing we'd want to do is label our equations 1 and 2, so that we know which is which.
And then the new equation we're forming is going to be equation 1 minus equation 2. So we'd do 7x minus 3x to get 4x, 2y minus 2y, which is 0, remember we wanted to cancel out the y's, and 23 minus 11, which is 12. So we end up with 4x equals 12, which we can then solve by dividing both sides by 4, to get x equals 3. And now that we've found the x value, all we have to do is plug that back into one of the original equations to find the corresponding y value. It's up to you which one you pick, so just use whichever one you think looks easiest. I'm going to go with this first one though.
So when I substitute in the x equals 3 to this equation, it will give 7 times 3 plus 2y equals 23. which simplifies to 21 plus 2y equals 23, then 2y equals 2, and finally y equals 1. So our solutions for this equation are x equals 3 and y equals 1. So if we just quickly think back to what we did for a second, remember that first of all we combined the equations to eliminate the y terms, then we used that combined equation to find the value of x, And then finally we use that value of x and one of the original equations to find the value of y. If you want to double check whether your answer is right, which is generally a good idea in an exam, you can try plugging both of your values into one of the original equations to check it all works. But because we plugged the x value into the first equation a minute ago when we tried to find our y, you should use the other equation this time. So in this case we should use the second equation.
So plugging in x is 3 and y is 1 would give us 3 times 3 plus 2 times 1, which should equal 11. So that simplifies to 9 plus 2, which does equal 11. So we know that our x and y values are correct, because if we had got the wrong x and y values, then it wouldn't have equaled 11. Okay, let's now try another example, which is a bit harder. and will involve us having to go through all of these steps one by one. So don't worry about them just now, we're going to look at them all in turn.
So in this question we're trying to solve the simultaneous equations 4x plus y equals 10 and 3y equals 2x minus 19. The first step is to label the two equations, 1 and 2, which will stop us from getting them mixed up later, and then make sure both of the equations are in the correct form. of something x plus something y equals some number. The first equation is already in this form so we don't need to do anything to it, but this second equation has the x term over here on the right rather than on the left. So to get it into the correct form that we want we're going to have to subtract 2x from both sides giving us negative 2x plus 3y equals negative 19. and we should also label this one as number 2 because it's the same equation.
Next we need to get the same number of x's or y's in both equations, because we need to have the same number of them before we can eliminate them. For example, because this first equation has 4 x's, we could try and make the second equation also have 4 x's as well, which we could do by multiplying the whole second equation by negative 2. to get positive 4x minus 6y equals 38. And again we label the equation 2 because it's the same equation. Now that we have the same number of x's in both equations we need to eliminate them, so get rid of the x terms.
To do this we can just do the entire first equation minus the entire second equation. So we just take away each term one by one. So 4x minus 4x which gives us 0, then 1y minus negative 6y, which is the same thing as adding 6y, so we'll get positive 7y, and finally 10 minus 38, which is negative 28. So if we get rid of the 0, then we're left with 7y equals negative 28. Then for step 4, we can solve this equation to find y by dividing both sides by 7. to find that y equals negative 4. And now that we have the y value, we just substitute that back into one of the original equations to find the corresponding x value.
So if we chose to substitute it into equation 1, we'd get 4x minus 4 equals 10, which we can simplify by adding 4 to both sides to get 4x equals 14, and then dividing by 4 to get x equals 3.5. And that's it. We now know that x is 3.5 and y is negative 4. So we now have our answer, but it's probably best to double check.
To do this, we can substitute the x and y values back into the other equation. So into equation 2, just to double check it works. So that would be 3 times negative 4 should equal 2 times 3.5 minus 19. so minus 12 equals 7 minus 19, which is correct because 7 minus 19 is negative 12, so we know that our x and y values were good. Hey everyone Amadeus here, I just wanted to let you know that we also have a learning platform where you can watch all of our videos, practice what you've learned with questions and keep track of all of your progress. for both the sciences and maths.
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