Video: Finding the Equation of a Line with Given 𝑥𝑦-Intercepts and Calculating the Area of a Triangle

Find the equation of the line with 𝑥-intercept 3 and 𝑦-intercept 7, and calculate the area of the triangle on this line and the two coordinate axes.

04:17

Video Transcript

Find the equation of the line with 𝑥-intercept three and 𝑦-intercept seven and calculate the area of the triangle on this line and the two coordinate axes.

So this question has two parts. We’re first asked to find the equation of a line and then we’re asked to calculate the area of this triangle. I think a diagram would be helpful here in order to visualize the situation. So we have a pair of coordinate axes. We’re told that this line has 𝑥-intercept three, which means it cuts the 𝑥-axis at three. We’re also told the line has 𝑦-intercept seven, so it cuts the 𝑦-axis at seven. By connecting these two points, I have the line that I’m looking to find the equation of and I can see the triangle that I’m asked to find the area of. It’s this triangle here.

So let’s start with the first part of this question, which asks to find the equation of this line. I’m going to do this using the slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. I can work out one of these two values straight away. Remember 𝑐 represents the 𝑦-intercept of the line. And I’m told in the question that this is equal to seven. So the equation of the line is 𝑦 equals 𝑚𝑥 plus seven. I now need to work out the slope of this line. And in order to do so, I need the coordinates of two points on the line. Well, I can use the coordinates of these points, the 𝑥-intercept and the 𝑦-intercept.

The slope of the line remember is calculated as the change in 𝑦 divided by the change in 𝑥. So looking at my diagram and using these two points, I’m gonna find the change in 𝑦 first of all. I can see that as I move from left to right across the diagram, the 𝑦-coordinate changes from seven to zero, which is a change of negative seven. It’s really important that you consider this change in 𝑦 as negative seven, not seven. The line is sloping downwards from left to right, and therefore it has a negative gradient. Now, let’s look at the change in 𝑥. I can see that as I move from left to right across this diagram, the 𝑥-coordinate changes from zero to three, which gives me a change in 𝑥 of positive three.

Now, I can substitute the change in 𝑦 and the change in 𝑥 into my calculation for the slope of this line. And we have that the slope of the line is equal to negative seven over three. 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 𝑚 into the equation. I have then that the equation of this line is 𝑦 equals negative seven over three 𝑥 plus seven. Now, sometimes you may be asked to give your answer in a slightly different format, for example, a format that doesn’t involve fractions. So you’d need to multiply the equation there by three, but as it hasn’t been specified here I’m going to leave my answer as it is now. So that’s the first part of the question completed.

The second part asked me to calculate the area of the triangle formed by this line and the two coordinate axes. Now from the diagram, we can see that this is a right-angled triangle because the 𝑥- and 𝑦-axes meet at a right angle. To find the area of a right-angled triangle, we need to multiply the base by the perpendicular height and then divide by two. So looking at the diagram, I can see that the base of this triangle is that measurement of three units. The height of the triangle is seven units. Now, we refer to this as negative seven when we’re calculating the slope of the line because the direction was important. For when we’re just looking at the length of that line in order to calculate an area, we’ll take its positive value of seven. So our calculation for the area is three multiplied by seven divided by two. And this gives us an answer of 10.5 square units for the area of this triangle.

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