Video: Finding the Equation of a Line through Two Points Giving Your Answer in a Specified Form

Find the equation of the line that passes through the points 𝐴 (βˆ’10, 2) and 𝐡 (0, 5), giving your answer in the form of π‘Žπ‘¦ + 𝑏π‘₯ + 𝑐 = 0.


Video Transcript

Find the equation of the line that passes through the points 𝐴: negative 10, two and 𝐡: zero, five, giving your answer in the form of π‘Žπ‘¦ plus 𝑏π‘₯ plus 𝑐 is equal to zero.

Now, we’ve been asked to find the equation of this straight line in a specific form, with all of the terms on the same side of the equation. But we’ll begin by using the general form of the equation of a straight line. That’s 𝑦 equals π‘šπ‘₯ plus 𝑏, where π‘š is the slope of the line and 𝑏 is its 𝑦-intercept.

Now, it’s important to be aware here that this value 𝑏 is not necessarily the same in our general equation and in the requested form. We’re told that the line passes through the point with coordinates zero, five. So the 𝑦-intercept of our line is zero, five. And we can work out straightaway that the value 𝑏 in our general form is five.

To work out the value of π‘š, we can use the formula change in 𝑦 over change in π‘₯. Subtracting the coordinates of 𝐴 from those of 𝐡, and we have five minus two over zero minus negative 10. We need to be really careful here because we are subtracting a negative value. A common mistake would be to just have zero minus 10 in the denominator. But that wouldn’t actually be subtracting the π‘₯-coordinate of 𝐴 from the π‘₯-coordinate of 𝐡.

Simplifying, we have three in the numerator. And in the denominator, zero minus negative 10 is zero plus 10, which is 10. So the slope of the line is three-tenths. Substituting the values of π‘š and 𝑏 into the equation of our straight line then, and we have 𝑦 equals three-tenths of π‘₯ plus five.

Now, this isn’t in the required form because we’re asked to collect all the terms on the same side of the equation. And although the question doesn’t explicitly say this, the values of π‘Ž, 𝑏, and 𝑐 that we use should be integers. We have a fraction of three-tenths. But we can eliminate this by multiplying each side by the denominator of 10. Doing so and remembering we need to multiply every term in the equation, including the constant term, by 10, we get 10𝑦 equals three π‘₯ plus 50. Grouping all the terms on the left-hand side by subtracting three π‘₯ and subtracting 50 gives the equation 10𝑦 minus three π‘₯ minus 50 equals zero, which is in the requested form.

Alternatively, we could’ve grouped the terms on the other side of the equation, which would give the equivalent form negative 10𝑦 plus three π‘₯ plus 50 equals zero, the exact negative of the equation we’ve given.

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