Video: Finding the Equation of a Line given Its Slope and a Point on the Line

A line 𝐿 has a slope of βˆ’2 and passes through the point (2, βˆ’3). Work out the equation of the line, giving your answer in the form 𝑦 = π‘šπ‘₯ + 𝑐.

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Video Transcript

A line 𝐿 has a slope of negative two and passes through the point two, negative three. Work out the equation of the line, giving your answer in the form 𝑦 equals π‘šπ‘₯ plus 𝑐.

So we’ve been given two pieces of information about this line. First of all, we’re told the slope of the line. We’re also told the coordinates of one point on the line. We’ve also been asked to give our answer in a particular form 𝑦 equals π‘šπ‘₯ plus 𝑐, which is slope-intercept form. Remember π‘š represents the slope of the line and 𝑐 represents the 𝑦-intercept. So what we need to do within this question is determine the values of π‘š and 𝑐.

We’ve actually been given the value of π‘š within the question itself as we’re told that it’s equal to two. This means that straightaway we can substitute the value of π‘š into the equation of the line. So we have that 𝑦 is equal to negative two π‘₯ plus 𝑐.

Now, we need to determine the value of 𝑐, the 𝑦-intercept of the line. To do this, we’ll use the coordinates of the point that we know lies on the line. This point satisfies the equation of the line, which means that when π‘₯ is equal to two, 𝑦 is equal to negative three within this equation. We can therefore substitute the values of π‘₯ and 𝑦 in order to find the value of 𝑐. So by substituting π‘₯ equals two and 𝑦 equals negative three, we now have the equation negative three is equal to negative two multiplied by two plus 𝑐.

And here is the equation that we can solve to find the value of 𝑐. Negative two multiplied by two is negative four. So now we have negative three is equal to negative four plus 𝑐. In order to determine 𝑐, I need to add four to both sides of the equation. And this tells me that 𝑐 is equal to one.

The final step is I need to substitute this value of 𝑐 into the equation of the line. So this gives us our answer to the problem. The equation of this line in slope-intercept form is 𝑦 is equal to negative two π‘₯ plus one.

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