Lesson video

Thank you for choosing to learn using this video today.

My name is Miss.

Davis and I'm gonna help you as you work your way through this lesson.

Feel free to pause the video and work through things at your own pace.

Lots of exciting things to look at, so let's get started.

Welcome to our lesson on solution to an equation.

By the end of the lesson, you will appreciate that a solution is a value that makes the two sides of an equation balance.

So we have a new keyword for today.

A solution to an equality with one variable is a value for the variable which, when substituted maintains the equality between the expressions.

And we're gonna explore what that means.

The process of finding a solution is called solving.

We're gonna have a look first at testing solutions to equations.

In an expression, the variable can take any value.

Sometimes this is restricted by context, so we might have values that have to be positive or they have to be integers.

We can use substitution to find the value of the expression for different values of the variable.

For example, what is the value of the expression 3P minus seven, when P is five, why don't you pause the video and give this one a go.

So what we can do is we can substitute five as the value of P.

When we substitute into an expression or an equation, we put the value in brackets.

So I've written this as three lots of five, subtract seven, then calculate.

15 minus seven equals eight.

This means that when P is five, 3P minus seven, has the value of eight.

We can find different values of the expression by substituting different values for the variable.

So we already worked out that when P is five, this expression has a value of eight.

So let's work out the value of this expression for other values of P.

Let's try when P equals negative three.

So three lots of negative three minus seven.

That's negative nine minus seven, which is negative 16.

I'm gonna fill that in in my table.

When P is negative two, we get negative six minus seven equals negative 13.

Let's fill that one in and we'll do the same when P is negative one, that gives us an answer of negative 10.

Okay, what I wanna know now is when does 3P minus seven have a value of two? If you think you know how to do it, pause the video and give it a go.

If not, I'll give you some help.

Okay, so what we need to do is fill in some other values for 3P minus seven.

Now this is a linear expression, so as the value of P increases by one, the expression increases at a constant rate.

You might have already spotted a pattern, so I can actually fill in the rest of my values.

It's worth checking that they work using substitution though.

I wanted to know when 3P minus seven has a value of two.

Well that's when P is three.

I can now see that in my table, when P is three, 3P minus seven equals two.

Let's try some other ones.

So if we now form an equation 3P minus seven equals five, we can find the specific value of that unknown that makes it true.

So 3P minus seven equals five is true.

When P equals four.

The solution to the equation, 3P minus seven equals five is when P has a value of four.

A solution to an equation with one variable is a value for the variable that makes both sides of the equation balanced.

So here we've got an equation with a variable of P.

So we're looking for a value for P that makes both sides of that equation balance.

The process of finding a solution is called solving.

So if you're asked to solve something, you're being asked to find the solution.

So what's the solution to 3P minus seven equals negative one.

Looking in our table, it's when P equals 2.

You can try this one, what is the solution to the equation 3P minus seven equals negative 10.

That has a solution when P equals negative one.

June says, "The solution to 3P minus seven equals 17 is when P has a value of eight." How could we check if June is correct? Right, we don't really want to carry on our table, but we can check his solution using substitution.

So let's substitute P is eight into our equation.

So we've got three lots of eight minus seven equals 17.

That gives us 24 minus seven equals 17, which gives us 17 equals 17.

Because both sides of that equation are balanced, P has a value of eight, is a solution to our equation.

Here is a table for the expression 4X plus five.

I've chosen some different values of X to substitute.

Notice that I haven't got a pattern with my X values.

I've just chosen some different values.

I've got 2, 1, 0, -2, -1.

I've also picked some non-integer values.

X does not have to be an integer and often solutions to equations will be non-integers.

Let's fill in these parts of our table.

So four multiply by half, add five, 2 add five which is seven, and four multiply by three over two plus five is 11.

We're gonna use our completed table now to find the solution to 4X plus five equals five.

There we go, it's in our table, it's when X is zero.

What is the solution to the equation 4X plus five equals one? Try this one yourself.

We need to find one in our table, and we're looking for when 4X plus five has a value of one.

So that's where we're looking.

So the solution is when X is negative one, right? What could the solution be to the equation 4X plus five equals zero? You might notice that that isn't in our table.

4X plus five does not have the value of zero anywhere in our table.

That means that none of those X values will be our solution.

What we can see is that zero is between negative three and one.

So that means the solution has to be between negative 2 and negative one.

Why don't we just try a number? Let's try negative 1.

5.

Substitute into our equation, four lots of negative 1.

5 plus five, negative six plus five, oh that's negative one.

We wanted zero, okay? It must be bigger than negative 1.

5, but smaller than negative one.

Let's try negative 1.

25.

Four lots of negative 1.

25 add five.

Negative five add five.

Oh, that does give us zero.

Lovely, so our solution is when X equals negative 1.

25.

You might be thinking at the moment, "That took us a lot of time and a lot of thinking." That's because substitution is not an efficient method to find a solution, and you will explore methods for finding a solution at a later date.

It can be time consuming when the solution is very big or very small and it can be difficult to find when not an integer.

However, where this method really comes into play is it's useful for checking a solution once you have found one.

We know that a solution makes both sides of the equation balance, so if we think we know what the solution is, we can test it out.

So which of these equations have a solution of X equals three? Give it a go and then we'll look at your answers.

Okay, when X equals three, two times three is six.

Six does not equal three, so not the top one.

When X equals three, 13 subtract six equals seven, or 13 subtract six is seven.

Seven equals seven, so that one is correct.

C, five lots of three is 15, subtract 12 is three.

Three is not equal to negative two.

So it's not a solution to the third one and three plus 17 equals 20.

3 plus 17 is 20.

20 equals 20, so it is a solution to the bottom one.

Well done if you found both of those.

Okay, we can use this method to test solutions when we have unknowns on both sides of our equation.

Could you use a balance to show this? So this represents the equation, 3X plus one equals 2X plus five.

Can you think of a value for X that will keep this scale balanced? Pause the video, spend a couple of minutes.

Can you find one? If not, don't worry, we'll look at it together.

I'm gonna try one.

I'm gonna try when X is five, when X is five, the left hand side of our balance will be 16.

The right hand side of our balance will only be 15, so it won't be balanced.

So it can't be when X is five.

Let's try when X is four then.

The left hand side of our balance is now 13.

The right hand side is also 13, so balanced.

Sophia says, "The solution then is 13." What mistake has she made? Well then this one's really important.

The solution is the value of X that made the equations balanced, not the value of the expression.

So the solution is when X is four 'cause that's the value of X that made the equation balanced.

Well done if you found the solution of when X is four by yourself.

Okay, so this table shows the value of five different expressions for five different values of X.

I'm gonna give you some equations.

I'd like you to find the solutions.

They're all in the table somewhere.

I'd like you to pause the video, see if you can work out where these solutions are, and then we'll come back together and see if you're right.

All right, well then if you did this, let's have a look together now at how we would find them.

So we're looking at 2X plus one equals three lots of X plus one.

So you are looking at those two rows there, and you're looking for when they are equal.

So they are equal when they both have a value of negative three.

However, the solution is the value of X that makes them equal.

So the solution is found when X equals negative two.

If you weren't sure about how to do this, you might wanna pause the video now and see if you can do the other three, then I'll go through the answers.

Okay, so the next one we're looking at when 2X equals seven minus 12X.

So find those two rows in your table and you're looking for a value when they're the same.

There you go, so they both equal one when X is a half.

The solution then is when X equals a half.

2X and three lots of X plus one has a solution when X equals negative three.

And the final one has a solution when X equals two thirds.

Okay, Izzy says, "The solution to 2X equals 2X plus one is X equals one." 'cause they're both the equations equal one.

You can see where she's highlighted the table.

Is she correct? Perfect, no, the expressions have to be equal for the same value of X.

Now, in fact, there is no solution to this equation and we're gonna have a look at why in a moment.

Some equations have no solution.

That's because there's no values of X that will make them true.

So there's no values of X which will make 2X the same as 2X plus one.

If you think about it, if you have a number and then you add one to it, it will become one bigger.

There's no way to make it stay the same by adding one.

Which of these then do you think will have no solution? See if you can find any patterns.

We are gonna explore this more with graphs in the next part of the lesson.

However, it's these three that have no solution.

You can't subtract six from a number and keep it the same.

You can't add four to the same number and keep it the same.

The third one, it does have a solution.

It has a solution when X is zero, 'cause zero plus eight is the same as two lots of zero plus eight.

They're both equal eight.

So that does have a solution when X is zero.

The bottom one if you expand the rackets is the same as 2X plus two equals 2X plus three.

You can't add two to a number and add three to the same number and get the same answer.

Can you find a solution for three lots of X minus two equals 3X minus six? You might wanna pause the video and explore this.

Try out a few values for X and see what happens.

Right, any value of X will balance the left and the right hand of this equation.

This is because those expressions are actually equivalent.

They are the same thing.

If you expand the left hand side, three lots of X minus two, you get 3X minus six.

So the left and the right are identical expressions.

They're always going to be true for any value of X.

When an equation is true for all values of the variable, it is called an identity.

So three dots of X minus two equals 3X minus six isn't actually an equation, it's an identity.

So it should really be written with an identity symbol.

It's always going to be the case that the left and the right hand side balance for any value of X.

That means there's an infinite number of solutions.

Okay, the solution to this equation is a positive integer less than six.

Let's see how we might find it.

So we can use substitution to find the solution.

We're gonna use a table to show our working out.

So those are my two expressions, and I've been told that it's a positive integer less than six.

Well let's try when X is one then.

So substituted X is one into both of my expressions.

They're not the same.

Let's try X equals two, five subtract three lots of two.

So five subtract six is negative one.

Two subtract 11 gives me negative nine.

When X equals three I get negative four and negative eight.

When X equals four I get negative seven and negative seven.

That's looking good, let's just try X equals five.

I get negative 10 and negative six.

So the solution to our equation is when those two things are balanced, and that is when X equals four.

So our solution is when X equals four.

A chance for a check then.

Which of these is the solution to 2A equals 3A plus five.

What do you think? Level A, you need to substitute in and see which one works.

Well then if you found that it was negative five.

Time for a practise.

I would like you to fill in the table for both expressions, and then use your table to find the solution.

It is gonna be one of the values that is in the table.

Give those a go and then come back.

Lovely.

Part two.

I would like you to sort these equations into the correct column in the table.

So some of these have a solution of X equals two.

Some of them have a solution of X equals zero, and some of them have no solution.

Can you put them in the correct place? Let's have a look at our answers.

So check your table is correct.

You should have found a solution when X equals four.

For B, you should have found a solution when X equals negative six.

And for C, you should have found a solution when X equals five over two.

Make sure you're happy with that before moving on.

Okay, lots of these had a solution of X equals 2, so 5X equals ten.

Five lots of X plus one equals 15, 3X minus six equals zero, and 2X minus two equals six minus 2X.

That might have taken quite some time to do.

That's absolutely fine, because you've got to substitute into both sides of the equations.

X equals zero, X equals 3X has a solution when x equals zero, X minus two equals 10X minus two has a solution when X is zero.

And finally those ones with no solution, 2X plus five equals 2X plus three, two minus X equals seven minus X, and three lots of 2X plus five equals 6X minus two.

Well done.

We're now gonna have a look at finding solutions from graphs.

If you have graphing software available, you might want to be using that as we work through this part of the lesson.

We can use graphs to find solutions to equations with unknowns on one side.

We're gonna find the solution to the equation 6X minus eight equals 10.

What we can do first is we can graph the value of the expression, 6X minus eight.

So there we go, I have used graphing software to speed things up, so that is the equation of Y equals 6X minus eight.

We've essentially graphed the expression 6X minus eight as X changes.

What we want to know is when this expression is equal to 10, so we can draw the graph of the equation Y equals 10 to help.

You may need to adjust the axes or zoom out to see both graphs.

I need to do that this time.

So now I've got the graph of the expression 6X minus eight, and the graph of Y equals 10.

Where can we see that 6X minus eight is equal to 10.

Pause the video, have a look at either my image or on your graphing software.

Right, it's where the graphs of the equations intersect.

The graphs intersect at (3,10).

So the solution to the equation 6X minus eight equals 10 is when x equals three.

We can check that.

Six lots of three minus eight equals 10, 18 minus eight equals 10, and that gives us 10 equals 10 which is balanced.

How could we find the solution to 6X minus eight equals negative two.

Pause the video and give it a go.

Lovely, well, I still have my expression of 6X minus eight graph, and we need to see where it intersects with Y equals negative two.

So I've drawn that line.

They intersect at (1,-2).

So the solution to the equation 6X minus eight equals negative two is found when X equals one.

How can we find the solution to 6X minus eight equals seven? Let's do this one together.

I want to know where they intersect, so I can draw the line Y equals seven and I'm looking for that coordinate.

Now if you're using graphing software, you can actually click on the point, and you'll be able to see that the solution is 2.

5.

If you're drawing it by hand, it can be quite tricky to see exactly where the line's intersect if it's not on an integer coordinate.

Okay, so line A, which is the pink one, has the equation Y equals three.

Line B, which is the green one, has the equation Y equals X plus five, line C, the purple one, has the equation Y equals 2X minus four, line D, that blue one, has the equation Y equals negative three.

What equations will we be able to find solutions for using these lines? What do you think? Lovely, we could have X plus five equals three, X plus five equals negative three, 2X minus four equals three and 2X minus four equals negative three.

You could also find 2X minus four equals X plus five, however that intersects off this picture so we wouldn't be able to use this picture.

Which of these equations will have negative solutions? How do you know? Lovely, it's X plus five equals three and X plus five equals negative three 'cause the lines intersect on a coordinate with a negative X value, and see it there on my graph.

Let's work out the solutions then to each of these equations.

First one, X equals negative two.

Second one, X equals negative eight.

Well X equals 3.

5 and X equals 0.

5.

Time for you to have a check to the line of equation Y equals 2X minus three has been drawn on the graph below.

It's that purple diagonal line.

I've also drawn three horizontal lines in to help you.

Use them to find the solution to those three equations.

Off you go.

well done, first one has solution when X equals two.

Second one is solution when X equals 3.

5, and the third one solution when X equals one.

This time the line with equation Y equals one minus X has been drawn.

Use the horizontal lines to find the solution.

Off you go.

Lovely, we have a solution when X equals zero, when x equals negative three, and when x equals two, well done.

We can now use a similar method to find solutions to equations with unknowns on both sides of the equals sign.

Let's look at 4X minus eight equals X plus one.

I'm gonna start by graphing the expression 4X minus eight, like I did before.

And we wanna know when this is equal to X plus one so we can also graph the expression of X plus one.

If we wanna know when this equation is balanced, we're looking for a value of X, which gives those two expressions the same value is there on our graph where they intersect.

So Aisha says, "The solution is (3,4), Jacob, "The solution is three." And Lucas, "The solution is four." What do you think? Jacob was correct.

The solution is the value of X which makes both expressions have the same value.

So this is when X is three.

They have the same value when X is three.

You can read it off your X-axis.

What does the number four represent then? What do you think? Right, four is the value of the expressions on both sides of the equation when X is three.

So they both have the same value, which is four, but it occurs when X equals three.

So that is our solution.

Let's try another one together, we're gonna do this manually, so we're gonna find the solution to six minus 2X equals 3X minus nine.

They're both linear, so both graphs will be straight lines.

So let's fill in a few values.

If you're not sure about filling in a table of values, pause the video and remind yourself how to do it.

Let's do the same for our other equation, and then we can plot those.

So plot the points and draw them accurately with a straight line using a ruler.

You also need to make sure your lines go across your entire axes.

Now we can see where they intercept.

So the solution is when X equals three.

They have a value of zero when X equals three.

But remember it's that X coordinate that we're looking for.

That is the solution.

We know that some equations do not have any solutions.

We saw those earlier in the lesson.

These ones will not have any solution.

Why do you think that is? Can you use your idea of graphs to come up with a reason? Right, they have the same coefficient of X on both sides of the equation.

We talked about this in the first part of the lesson by saying that means you're trying to add a different value onto the same number and get the same answer, which is not possible.

We're now gonna link that to graphs.

If I plot the expressions X and X minus six, they look like this.

The lines are gonna be parallel.

They have the same coefficient of X, so they're gonna have the same gradient.

So that means they're gonna be parallel.

Parallel lines never cross, so they will not have a solution.

When the equation of a line is written in the form Y equal MX plus C, the coefficient of X is the gradient.

You might have come across this in your graphing skills.

When both lines have the same gradient, they'll be parallel and they'll never intersect.

Lucas says, "The equation 3X plus four equals 3X minus six does not have a solution." Is Lucas correct? How do you know? Yes, he is correct.

The graphs of the equation Y equals 3X plus four, and Y equals 3X minus six will both have a gradient of three because the coefficient of X is three.

Therefore, they will be parallel and never intersect.

There will never be a value of X which will balance this equation.

Have a look at your answer, see whether it contains those elements.

Fantastic, so below are the graphs of the equations Y equals three minus 4X, Y equals two lots of 2X plus 2, and Y equals 4X minus three.

They've been labelled A, B, and C.

So make sure you're happy with which one's which.

Which of these equations will have solutions? What do you think? Perfect, the top one will, it has a solution when X is 0.

75.

The second one will, has a solution when X equals negative 0.

125, C are parallel, so no solution.

If you expand the bracket, you can see that that's 4X plus four.

They both have the same coefficient of X, so the same gradient.

Time for you to put this all into practise.

For question 1A, I'd like you to use the lines to solve those equations.

You have two different equations to solve.

The names of the lines being given for you.

For B, I've drawn the graph of the equation Y equals 5X minus three, it's the green one.

I'd like you to use it to solve the two equations underneath.

And C, solve the equations below by drawing any relevant graphs.

Off you go, and then we'll look at the next set.

Well done, so for two, I've drawn three lines on the graph for you and I've labelled them.

I'd like you to use them to solve those three equations, A, B, and C.

And finally, you are gonna do it all yourself.

You're gonna solve the equation by drawing the relevant graphs.

So you're looking to solve the equation, 3X minus three equals five minus X.

I've given you some tables of values that you might want to use to help you, and then find the solution.

Come back when you're ready for the answers.

Well done, so for the first one, you have a solution when X is six.

The second one, a solution when X is two.

For B, you have a solution when X equals zero, well done if you've got that one.

And then a solution when X equals negative one.

And for C, you might have wanted to draw in the line Y equals three and y equals one.

And then you've got a solution when X is one, and a solution when X is zero for that second equation.

And then for two, you are looking at where the line Y equals X, and the line Y equals three minus 2X intersect, and that's when X is one.

For B, you're looking at the intersection between Y equals X and Y equals 3X plus eight, and that's when X is negative four.

And for C, you are looking for where the line Y equals 3X plus eight intersects with y equals three minus 2X and that's when X equals negative one.

And finally, check your tables of values are the same as mine.

And then if you draw both lines in accurately using a ruler, you'll get an intersection point when X equals two.

If you are plotting of your coordinates weren't that accurate, you might have found that it doesn't intersect exactly at X equals two.

However, you can see in the tables that they both have a value of three when X has a value of two, which is helping us as well see that solution.

Fantastic work today.

We have seen that in a linear equation there is a particular value that makes the equation balance, and we're using this language of a solution.

Finding a solution is called solving.

We can test whether a value is a solution by substituting it into the equation, and we can find a solution to an equation by graphing the expressions on both sides, and seeing where they intersect.

I'm really glad you decided to join us today, and I really hope you choose to learn with us again.