WEBVTT
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In this video, weβre going to look at how to find the equation of a straight line in various different forms, given two pieces of information, the slope of the line and the π¦-intercept.
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Firstly, a reminder of the different formats that you may be asked to give the equation of a line in.
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The first is slope-intercept form, π¦ equals ππ₯ plus π.
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The letters π and π each represent particular properties of the straight line.
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π represents the slope of the line, which means, for every one unit you move to the right, the line moves this many units either up or down, depending on whether π is positive or negative.
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π represents the π¦-intercept of the line, which is the value at which the line cuts the π¦-axis.
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The second common format for the equation of a straight line is point-slope form, π¦ minus π¦ one equals π π₯ minus π₯ one.
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π represents the slope of the line, as we saw previously.
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π₯ one, π¦ one represents the coordinates of any particular point that lies on this given straight line.
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Now, in this video, weβre looking at finding the equation of a line given its slope and its π¦-intercept.
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And therefore, itβs usual that we would be using the slope-intercept form in order to do this.
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Determine, in slope-intercept form, the equation of the line which has a slope of eight and a π¦-intercept of negative four.
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So weβre told the way that our answer should be expressed, slope-intercept form, π¦ equals ππ₯ plus π.
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So what we need to do is determine the values of π and π for this question.
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Weβre actually given the values of π and π explicitly within the information in the question itself.
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Weβre told that this line has a slope of eight; this means that the value of π is eight.
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Weβre also told that the line has a π¦-intercept of negative four; this means that the value of π is negative four.
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So all we need to do to answer this question is substitute the values of eight and negative four into the slope-intercept form of a straight line.
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Therefore, we have that the equation of the straight line is π¦ equals eight π₯ minus four.
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Find the coordinates of the point where π¦ equals four π₯ plus 12 intersects the π¦-axis.
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So in this question, weβre given the equation of a straight line in slope-intercept form.
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And weβre asked for the coordinates of the point where it intersects the π¦-axis.
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What this question is checking then is do we understand slope-intercept form and what the different parts of the equation represent.
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Remember that slope-intercept form is π¦ equals ππ₯ plus π, where π represents the π¦-intercept of the straight line, which is what weβre looking for here.
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The π¦-intercept is the point where the line intersects the π¦-axis.
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So we can see by comparing the general form and the specific straight line that we have, the value of π here is 12.
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But the question doesnβt just ask us for the value of π; it asks us for the coordinates of this point.
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So the π¦-intercept remember is a point on the π¦-axis.
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Weβve just worked out its π¦-coordinate; itβs this value of 12.
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To work out the π₯-coordinate, we just need to remember that at every point on the π¦-axis the π₯-coordinate is zero.
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You could perhaps see this more clearly by picturing what the graph would look like as Iβve done here.
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So the coordinates of this point then are going to be zero, 12.
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And that is our final answer to this question.
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Write the equation represented by the graph shown.
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Give your answer in the form π¦ equals ππ₯ plus π.
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So we have a diagram of a straight line.
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And weβre asked to give its equation in slope-intercept form, which means we need to work out what each of these two things are.
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From looking at the diagram, we can see that the π¦-intercept is negative four, which means that the value of π, which is the letter used here to represent the π¦-intercept, must be negative four.
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So I can write down the beginnings of the equation of this straight line; itβs π¦ equals ππ₯ minus four.
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Next, we need to find the value of π, the slope of this line.
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And in order to do this, I need the coordinates of two points that lie on the line.
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Weβve already had identified one point, the point with coordinates zero, negative four.
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Looking at the graph, I can also see that thereβs a point here that would be convenient to use.
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This point lies on the π₯-axis and has the coordinates six, zero.
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So Iβm going to use these two points to calculate the slope of the line.
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So, the slope of the line can be calculated as a change in π¦ divided by a change in π₯.
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Or you can think of this as π¦ two minus π¦ one over π₯ two minus π₯ one, if you choose to label the two points as π₯ one, π¦ one and π₯ two, π¦ two.
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Iβm just going to look at the diagram in order to work out the change in π¦ and the change in π₯.
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The change in π¦ first of all then, well that is the vertical length in this triangle.
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And I can see that it moves from a π¦-coordinate of negative four to a π¦-coordinate of zero.
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Therefore, the change in π¦ is positive four.
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Now, letβs look at the change in π₯; this is the horizontal change.
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So I can see from the diagram that this moves from a value of zero to a value of six, which gives me a change in π₯ of positive six.
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So the slope of this line then is four over six.
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But this can be written as a simplified fraction; itβs two-thirds.
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Finally then, I just need to substitute this value of π, the slope of the line, into the equation.
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So the equation of the line represented by this graph is π¦ equals two-thirds π₯ minus four.
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Find the equation of the line with π₯-intercept three and π¦-intercept seven and calculate the area of the triangle on this line and the two coordinate axes.
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So this question has two parts.
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Weβre first asked to find the equation of a line and then weβre asked to calculate the area of this triangle.
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I think a diagram would be helpful here in order to visualize the situation.
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So we have a pair of coordinate axes.
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Weβre told that this line has π₯-intercept three, which means it cuts the π₯-axis at three.
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Weβre also told the line has π¦-intercept seven, so it cuts the π¦-axis at seven.
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By connecting these two points, I have the line that Iβm looking to find the equation of and I can see the triangle that Iβm asked to find the area of.
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Itβs this triangle here.
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So letβs start with the first part of this question, which asks to find the equation of this line.
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Iβm going to do this using the slope-intercept form, π¦ equals ππ₯ plus π.
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I can work out one of these two values straight away.
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Remember π represents the π¦-intercept of the line.
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And Iβm told in the question that this is equal to seven.
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So the equation of the line is π¦ equals ππ₯ plus seven.
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I now need to work out the slope of this line.
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And in order to do so, I need the coordinates of two points on the line.
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Well, I can use the coordinates of these points, the π₯-intercept and the π¦-intercept.
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The slope of the line remember is calculated as the change in π¦ divided by the change in π₯.
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So looking at my diagram and using these two points, Iβm gonna find the change in π¦ first of all.
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I can see that as I move from left to right across the diagram, the π¦-coordinate changes from seven to zero, which is a change of negative seven.
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Itβs really important that you consider this change in π¦ as negative seven, not seven.
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The line is sloping downwards from left to right, and therefore it has a negative gradient.
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Now, letβs look at the change in π₯.
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I can see that as I move from left to right across this diagram, the π₯-coordinate changes from zero to three, which gives me a change in π₯ of positive three.
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Now, I can substitute the change in π¦ and the change in π₯ into my calculation for the slope of this line.
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And we have that the slope of the line is equal to negative seven over three.
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Finally, in order to complete the first part of the question and find the equation of the line, I need to substitute this value for π into the equation.
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I have then that the equation of this line is π¦ equals negative seven over three π₯ plus seven.
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Now, sometimes you may be asked to give your answer in a slightly different format, for example, a format that doesnβt involve fractions.
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So youβd need to multiply the equation there by three, but as it hasnβt been specified here Iβm going to leave my answer as it is now.
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So thatβs the first part of the question completed.
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The second part asked me to calculate the area of the triangle formed by this line and the two coordinate axes.
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Now from the diagram, we can see that this is a right-angled triangle because the π₯- and π¦-axes meet at a right angle.
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To find the area of a right-angled triangle, we need to multiply the base by the perpendicular height and then divide by two.
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So looking at the diagram, I can see that the base of this triangle is that measurement of three units.
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The height of the triangle is seven units.
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Now, we refer to this as negative seven when weβre calculating the slope of the line because the direction was important.
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For when weβre just looking at the length of that line in order to calculate an area, weβll take its positive value of seven.
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So our calculation for the area is three multiplied by seven divided by two.
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And this gives us an answer of 10.5 square units for the area of this triangle.
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In summary then, when weβre given the slope and the π¦-intercept of a straight line, itβs usual to calculate its equation in slope-intercept form, π¦ equals ππ₯ plus π because the values can be just substituted directly into this form.
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We may need to calculate the slope of the line ourselves given two points on the line by using change in π¦ divided by change in π₯.
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It would also be possible to give the equation of a line in point-slope form, but this would be unnecessarily complicated if the information weβre given is slope and π¦-intercept as this aligns so conveniently with the slope-intercept form.