### Video Transcript

Consider the line with equation two ๐ฅ plus two ๐ฆ equals four. The points ๐พ and ๐ฟ lie on the line. Part a) Complete the coordinates of ๐พ and ๐ฟ. Part b) Hence or otherwise, draw the line two ๐ฅ plus two ๐ฆ equals four for the values of ๐ฅ from negative three to three.

So weโve been given the equation of this line. Two ๐ฅ plus two ๐ฆ equals four. And weโre told that the points ๐พ and ๐ฟ lie on the line. Now if a point lies on a line, then it means that the coordinates of that point satisfy the equation of that line. So for any point on this line, it is the case that two multiplied by its ๐ฅ-coordinate plus two multiplied by its ๐ฆ-coordinate is always equal to four.

Letโs consider the point ๐พ first of all. We know that its ๐ฅ-coordinate is zero. But we donโt know its ๐ฆ-coordinate. So we can just call it ๐ฆ. As ๐พ lies on the line, we can substitute its coordinates into the equation of the line. Two ๐ฅ becomes two multiplied by zero. Two ๐ฆ is just still two ๐ฆs. We donโt know the value of ๐ฆ. And this is equal to four.

We now have an equation that we can solve in order to find the value of ๐ฆ. Two multiplied by zero is just zero. So our equation becomes two ๐ฆ is equal to four. To solve for ๐ฆ, we need to divide both sides of this equation by two. Two ๐ฆ divided by two is ๐ฆ, and four divided by two is two. So we have that ๐ฆ is equal to two. And therefore, the ๐ฆ-coordinate of the point ๐พ is two.

For the point ๐ฟ, we know that the ๐ฆ-coordinate is zero. And itโs the ๐ฅ-coordinate that weโre looking to find this time. We can do this in exactly the same way. Two ๐ฅ is still two ๐ฅ. And then two ๐ฆ becomes two multiplied by zero. So we have two ๐ฅ plus two multiplied by zero is equal to four. Two multiplied by zero is still zero. So we have two ๐ฅ equals four. And to solve for ๐ฅ, we need to divide both sides of the equation by two, giving ๐ฅ equals two. The coordinates of the point ๐ฟ then are two, zero.

Now letโs think about how this helps us with part b, which is asking us to draw the line two ๐ฅ plus two ๐ฆ equals four for values of ๐ฅ between negative three and three. We know that this is a straight line because there is a linear relationship between ๐ฅ and ๐ฆ. The only power of ๐ฅ and ๐ฆ that appears in the equation is one. We donโt have any ๐ฅ squareds or ๐ฆ squared terms, for example.

We also know the coordinates of two points that lie on the line. We have ๐พ, which is the point zero, two. We can plot this. Zero on the ๐ฅ-axis and two on the ๐ฆ-axis gives us the point here. We also know that the point ๐ฟ lies on our line, which has coordinates two, zero. So we go two units to the right of the origin and no units up. And it gives us the second point, here.

As this is a straight line, we only need two points in order to be able to draw it. We can now take our ruler and connect the two points that weโve drawn and then extend this line. However, itโs always good to have at least three points to make sure we havenโt made any mistakes.

We can choose any ๐ฅ-value that we like, or indeed any ๐ฆ-value, to work out the corresponding ๐ฆ- or ๐ฅ-value. Letโs choose ๐ฅ equals negative three. We substitute into the equation of the line in exactly the same way as we did in part a.

We have two multiplied by ๐ฅ โ thatโs two multiplied by negative three โ plus two ๐ฆ is equal to four. Two multiplied by negative three is negative six. Remember, positive multiplied by a negative gives a negative answer. So we have negative six plus two ๐ฆ is equal to four.

To solve for ๐ฆ, our next step is to add six to each side of the equation, giving two ๐ฆ is equal to 10. We can then divide both sides of the equation by two to give ๐ฆ is equal to five. This means that the third point on our line has the coordinates negative three, five. And so we can also plot this point.

We now take our ruler and connect the three points weโve drawn with a straight line, noting that they do indeed all lie on the same line. As weโve been asked to draw the line for ๐ฅ-values from negative three to three, we must make sure that our line reaches at least these values. But we can extend our line beyond these points if we wish.

The word โhenceโ in a question like this means we need to use the work weโve just done. So we use the coordinates of the points ๐พ and ๐ฟ, which weโd worked out in part a, to help us draw this line.

The word โotherwiseโ suggests that we could actually do this a different way. What we could do is rearrange the equation of our straight line into the general form ๐ฆ equals ๐๐ฅ plus ๐. Here ๐ gives the gradient of the straight line and ๐ gives the ๐ฆ-intercept, the value at which the line crosses the ๐ฆ-axis. We could use these two key features in order to draw the line.

Our line has equation two ๐ฅ plus two ๐ฆ is equal to four. We would first subtract two ๐ฅ from each side to give two ๐ฆ equals negative two ๐ฅ plus four and then divide the equation by two to give ๐ฆ equals negative ๐ฅ plus two. Comparing this with the general equation of a straight line, we see that the gradient of our line is negative one, because the coefficient or value in front of the ๐ฅ is negative one, and the ๐ฆ-intercept is positive two.

The ๐ฆ-intercept is this point that Iโve now marked in pink on our graph, the point where the line crosses the ๐ฆ-axis. And we can see that it is indeed two on the line that weโve drawn. The gradient of the line tells us, for every one unit we move to the right, how many units the line moves up or down. So the gradient of negative one means that, for every one unit we move to the right, the line goes down by one unit, which we can see is indeed the case on the line weโve drawn.

Our alternative method then wouldโve been to draw in the ๐ฆ-intercept at two and then draw some other points by moving one unit right and one unit down each time. We could then connect these points with a straight line.

So weโve completed the question. In part a, we found that the coordinates of ๐พ and ๐ฟ was zero, two and two, zero, respectively. And then in part b, weโve discussed two possible methods for drawing the line with equation two ๐ฅ plus two ๐ฆ is equal to four.