In this video, we are going to be looking at linear equations in one variable. Now first of all, let’s just make sure that we understand what’s meant by the different parts of this title. So first of all, one variable, this just means that the equations we’re gonna look at are only going to involve one unknown letter, so they might involve 𝑥 or 𝑦 or 𝑎, but there’s only going to be one letter involved in each of the equations that we look at.
The word linear means that the power of this variable, whatever it might be, is never going to be more than one, so our equations might involve for example the letter 𝑥 and they might involve constant terms like plus three, but they won’t involve terms like 𝑥 squared or 𝑥 cubed or anything that is more than just a simple 𝑥.
So here are some examples of the type of equations we might look at. All of them just involve one letter whether it’s 𝑎, 𝑚, 𝑥, or 𝑦, and the highest power of that letter is always just one.
Now we’re going to look at the different types of linear equation that you might encounter, and there are three different types that we’re going to look at.
The first type we’re going to look at are equations that have a unique solution. And what this means is that whatever the variable in the equation is, so suppose it’s 𝑥, there is only one single unique value of 𝑥 that satisfies that particular equation.
So we’ll see what we mean by this in the context of an example. So the equation we’re going to look at is this: five 𝑥 plus three is equal to eighteen.
Now looking at that equation, you may be able to determine the value of 𝑥 directly, but let’s think about a formal way to solve this equation. We want to work out the value of 𝑥 that works in this equation. So the first thing we rea- need to do is to look at this plus three term. And in order to eliminate that plus three term, I would have to do the opposite, which is subtracting three. So I would have to do that to both sides of this equation. So on both sides of the equation, I would need to subtract three. And if I were to do that the next line of my working out, well if I’ve subtracted three, then I just have five 𝑥 left on the left-hand side, and on the right-hand side if I’ve subtracted three from eighteen, I just have fifteen left.
The next step well I’ve got five 𝑥 and I just want one 𝑥, so I would need to divide both sides of this equation by five. And so if I divide five 𝑥 by five, I get 𝑥. And if I divide the other side by five, fifteen divided by five, I’m left with three.
So what I have here is I have a unique solution. It tells me that the value of 𝑥 must be equal to three, and three is the only value of 𝑥 that will work in this equation. If I go back to the top and try to substitute in any other value of 𝑥, it won’t work. I won’t get eighteen when I multiply it by five and add three. So this is the first type of equation, those that have unique solutions where only one single value of 𝑥, or whatever letter it might be, actually works in the equation. And these are all equations that can be reduced to a particular type; they can be reduced to something of the form 𝑥 is equal to 𝑎, where 𝑎 is just a constant, a number, so in our case 𝑥 is equal to three.
Second type of linear equation is one that has infinitely many solutions, which means in fact any value of the variable will work, doesn’t matter what you choose it to be; any single value that you choose will work within the equation. So again, let’s look at it in the context of an example.
So here I have an equation and I’m suggesting to you that in fact it has infinitely many solutions. So in order to see that the first step, well we might want to expand the bracket on the right-hand side of the equation.
So if I go ahead and do that, I have four 𝑥 plus seven is equal to four 𝑥 plus eight minus one. Now the next step I can do a bit of simplification on the right-hand side. I’ve got plus eight minus one, so that’s gonna simplify to plus seven.
So I’m left with this: four 𝑥 plus seven is equal to four 𝑥 plus seven. Now that is a statement that is always true, doesn’t matter what value 𝑥 takes whether it’s three or minus two or something more complicated, 𝜋. This will always be true for any value of 𝑥; four 𝑥 plus seven will always be equal to four 𝑥 plus seven because they’re the same expression.
If I do want to go a step further though, I can subtract four 𝑥 from both sides. And if I do that, I’m left with seven is equal to seven, which of course is true.
So this type of equation, infinitely many solutions, it’s the type of equation where it can be reduced to what we describe as 𝑎 is equal to 𝑎, so some number, some constant, is equal to itself. And in our case, seven is equal to seven. And if you can reduce an equation to a statement like that, then you know it has infinitely many solutions.
The third and final type is an equation which has no solution, and again we’ll look at this in the context of an example. So I have the equation two 𝑥 plus three is equal to two multiplied by 𝑥 minus one. Now again, the first step here might be to expand the bracket on the right-hand side.
And if I do that, I have two 𝑥 plus three is equal to two 𝑥 minus two. Now you may be able to see straight away that that doesn’t work, that doesn’t make sense, because if I have a value of 𝑥 and I double it and add three, how can I possibly get the same result as if I double it and subtract two? But if I want to see this a little bit more clearly, I can subtract two 𝑥 from each side of this equation.
And if I do that, I’m left with the statement three is equal to negative two. Now clearly, that statement is nonsense. Three is not equal to negative two, and so this equation has led to a contradiction. And for that reason, there are no solutions; there are no values of 𝑥 that will work in the original equation because it leads to this contradiction.
So this final type this is what’s known as an equation where we end up with the result 𝑎 is equal to 𝑏, where 𝑎 and 𝑏 are different numbers, so in our case three and negative two, but of course that can’t be the case.
So these are the three different types of linear equation that you’ll come across: one that has a unique solution, one that has infinitely many solutions, and one that has no solution. And you’ll know which situation you’re in depending on which of these three forms you can reduce it to. If you can reduce it to 𝑥 equals 𝑎, we have a unique solution; if you can reduce it to 𝑎 equals 𝑎, we have infinitely many solutions; and if you woul- can reduce it to 𝑎 equals 𝑏, a contradiction because the numbers are different, then there are no solutions to the original equation.
So let’s look at an example. I’m gonna begin with this: how many solutions are there to the equation five 𝑚 minus three is equal to three 𝑚 plus seven plus two 𝑚 minus ten? So we need to work out which of the three situations we saw on the last slide we are in in this case.
So let’s look at how we can simplify this equation. Now there’s not much I can do on the left-hand side, so I’ll leave it as five 𝑚 subtract three, but there are things I can do on the right-hand side. I have three 𝑚 plus two 𝑚, so that gives me five 𝑚 overall, and then I have positive seven subtract ten, so that gives me negative three overall.
So I’ve taken the equation and I’ve reduced it to the statement five 𝑚 subtract three is equal to five 𝑚 subtract three. Now both of those sides are the same, so you may be able to conclude at this stage that there are infinitely many solutions to this equation. But if we just continue so that we can get it into one of those final forms we discussed on the last page, we might want to subtract five 𝑚 from each side of this equation. And if I do that, I will be left with negative three is equal to negative three, so what we can see is we’re in that situation where we had infinitely many solutions; we’re in the situation where we reduced the equation to 𝑎 is equal to 𝑎. In our case, 𝑎 is negative three.
So we conclude that there are infinitely many solutions to this equation, which means it doesn’t matter what the value of 𝑚 is; any value of 𝑚 that I choose, if I substitute it into the original equation, it will work; it will hold true.
Let’s look at a second example. So how many solutions are there to the equation four lots of 𝑦 minus one plus two lots of 𝑦 minus three is equal to six 𝑦 plus four? So we want to simplify this equation a bit, and the first step would be to expand both of the brackets on the left-hand side.
So if I do that carefully, I have four 𝑦 minus four plus two 𝑦 minus six is equal to six 𝑦 plus four. Now we need to go a couple of steps further, so we need to simplify this by collecting the like terms together. So we have four 𝑦 plus two 𝑦, meaning I have six 𝑦 overall, and then we have negative four take away six, meaning I have negative ten overall. And the right-hand side of the equation stays as it was, six 𝑦 plus four.
So now we have the statement six 𝑦 take away ten is equal to six 𝑦 add four. If we carry on, if we subtract six 𝑦 from both sides of this equation, now we have the statement negative ten is equal to four. And at this stage here, we can see well that’s a contradiction; negative ten is not equal to four, and so what this tells us is we are in that third situation that we looked at before where we have two numbers that are not equal to each other. And therefore the equation doesn’t work; it doesn’t hold.
So what does that tell us? Well it tells us that we’re in that situation where we have 𝑎 is equal to 𝑏, and 𝑎 and 𝑏 are not the same, and therefore there are no solutions to this equation.
So there isn’t a single value of 𝑦 that we can substitute into the original equation and get it to work because we’ve got this contradiction: negative ten is equal to four. Now let’s look at a different type of question. So this question asked me to create an equation with infinitely many solutions, so I need to create one myself.
Now this means you must be in that second scenario, which means it’s an equation that can be reduced to 𝑎 is equal to 𝑎, where 𝑎 is some constant. So I’m gonna start with that and I’m gonna start with choosing 𝑎 to be five for example, and so I’m gonna start with five is equal to five.
Now that’s not a particularly complicated equation yet, so I want to do some other things to it. And the key point is, as long as I do the same thing to both sides of this equation, I will be able to build it up to a more complicated equation that still has infinitely many solutions.
So the first thing I’m gonna do is I’m gonna add four 𝑏 to both sides of my equation. So now I have four 𝑏 plus five is equal to four 𝑏 plus five. Still not a particularly complicated equation, so I’m gonna try and change parts of this. On the left-hand side, I’m gonna change that four 𝑏 for three 𝑏 add 𝑏.
So now I have three 𝑏 add 𝑏 plus five is equal to four 𝑏 plus five. And I would like to change something on the right-hand side, and I’m gonna change that five to four add one.
So now I have the equation three 𝑏 plus 𝑏 plus five equals four 𝑏 plus four plus one. It’s still exactly the same equation, but I’ve just changed parts of it as I go along. Like to do maybe one more thing by looking at this term or this part here, four 𝑏 plus four, and what I’m gonna do is I’m gonna take out a common factor of four from both of those terms so that I can write it as four multiplied by a bracket. So the left-hand side is gonna stay the same.
But on the right-hand side, I’m gonna take out this common factor of four. Now in the bracket, I will have 𝑏 plus one so that when I expand that bracket, I still have four 𝑏 plus four, and I still need this plus one outside the bracket.
So now I have a much more complicated looking equation than the one I started off with, but it is still exactly the same equation.
I started with a statement I wanted to end up with, five is equal to five, and then I just made sure I did the same thing to both sides, so I introduced this extra four 𝑏 to both sides. Then I manipulated it differently, but I never added or subtracted anything from one side that I didn’t do to the other. I just wrote it in a slightly different format, and it’s left w- me with a very different looking equation, but one that still has infinitely many solutions.
One final example, let’s look at creating an equation with a unique solution. So my equation is to only work for one particular value of the variable, so I’m gonna choose that the letter I’m gonna use in this equation is the letter 𝑝, and I’m gonna choose that the solution to my equation I would like it to be negative three. So I’m gonna start with 𝑝 is equal to negative three.
Now I want to build up my equation, so again I need to make sure I do exactly the same thing to both sides of this equation. So first of all, I am going to double both sides of my equation. And if I do that, well if I double 𝑝, I will have two 𝑝, and if I double negative three, I will have negative six.
So now I have the equation two 𝑝 is equal to negative six. Now I would like to do something else; this- so I would like to subtract four from both sides of the equation. So on the left-hand side, I will have two 𝑝 subtract four, and on the right-hand side, well if I’ve got negative six and I subtract four, now I’m going to have negative ten.
Next, I think I’d like to write the two 𝑝 differently, and I’d like to think of it as five 𝑝 minus three 𝑝, so I’m just gonna replace two 𝑝 with five 𝑝 minus three 𝑝. So now I have five 𝑝 minus three 𝑝 minus four equals minus ten.
The final thing, well I think I’d like to have 𝑝 on both sides of this equation. So I’d like to move this negative three 𝑝. I’d like to have it over on this side of the equation here. So in order to achieve that, I need to add three 𝑝 to both sides of this equation.
So if I add three 𝑝 to both sides, well on the left-hand side, I’ll now have five 𝑝 subtract four, and on the right-hand side, I’ll have three 𝑝 subtract ten.
And I’ll finish there for this equation. So what I did I started off with a statement, I started with 𝑝 is equal to negative three, and then I just built up from there. I did the same thing to both sides, so doubling, subtracting four, adding three 𝑝, but I’ve built up a much more complicated looking equation than the one I initially started off with. But it’s still exactly the same equation. And if I were start here and solve, I would end up with that same solution: 𝑝 is equal to negative three.
So there you have a summary of the three different types of linear equation that you might encounter, how you can determine which type of equation you have, and how you can work backwards to create an equation that either has a unique solution, infinitely many solutions, or no solutions.