Video: Finding the Equation of a Straight Line

Determine the equation of the line passing through ๐ด (0, 16) and ๐ต (1, โˆ’9) in slope-intercept form.

02:59

Video Transcript

Determine the equation of the line passing through ๐ด: zero, 16 and ๐ต: one, negative nine in slope-intercept form.

So in this question, weโ€™re given the coordinates of two points that lie on a straight line, and weโ€™re asked to determine its equation. Weโ€™ve also been asked for the equation in a specific form, slope-intercept form. This is ๐‘ฆ equals ๐‘š๐‘ฅ plus ๐‘, where ๐‘š represents the slope of the line and ๐‘ represents the ๐‘ฆ-intercept. In order to answer this question, we need to determine both of these values, the slope and the ๐‘ฆ-intercept.

So letโ€™s look carefully at the information in the question and the coordinates of the two points weโ€™ve been given that lie on the line. The first of these two points, ๐ด, has coordinates zero, 16. As the ๐‘ฅ-coordinate of this point is zero, this is a point on the ๐‘ฆ-axis. And therefore, these are the coordinates of the ๐‘ฆ-intercept. The ๐‘ฆ-coordinate is 16 and this tells us the value of ๐‘ in our equation is also 16. So we can substitute this value of ๐‘ into the equation of our line. So we have that ๐‘ฆ equals ๐‘š๐‘ฅ plus 16 for a value of ๐‘š that we now need to calculate.

Remember, ๐‘š represents the slope of the line. If we know the coordinates of two points on the line โ€” which weโ€™ll refer to as ๐‘ฅ one, ๐‘ฆ one and ๐‘ฅ two, ๐‘ฆ two โ€” then the slope can be calculated using this formula: ๐‘š is equal to ๐‘ฆ two minus ๐‘ฆ one over ๐‘ฅ two minus ๐‘ฅ one. It doesnโ€™t matter which of the two points we think of as ๐‘ฅ one, ๐‘ฆ one and which we think of as ๐‘ฅ two, ๐‘ฆ two. Iโ€™m going to choose to subtract the coordinates of point ๐ด from those of point ๐ต.

So letโ€™s calculate the slope of this line. ๐‘š is equal to ๐‘ฆ two minus ๐‘ฆ one, so that is negative nine minus 16. And then I need to divide by ๐‘ฅ two minus ๐‘ฅ one, so thatโ€™s one minus zero. My calculation for the slope simplifies to negative 25 over one, and of course thatโ€™s just equal to negative 25. The final step in answering this question is I need to substitute this calculated value of ๐‘š into the equation of the line.

So in doing so, I have that ๐‘ฆ is equal to negative 25๐‘ฅ plus 16. And thatโ€™s our answer to this problem, the equation of the line in slope-intercept form.

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