Question Video: Determining in Slope-Intercept Form the Equation of a Line Mathematics

Determine, in slope–intercept form, the equation of the line passing through 𝐴(13, −7) perpendicular to the line passing through 𝐵(8, −9) and 𝐶(−8, 10).

03:37

Video Transcript

Determine, in slope–intercept form, the equation of the line passing through 𝐴: 13, negative seven perpendicular to the line passing through 𝐵: eight, negative nine and 𝐶: negative eight, 10.

So, here’s what we’re thinking. We have points 𝐵 and 𝐶, which form a line. Point 𝐴 does not fall on this line. But point 𝐴 falls on a line perpendicular to the line 𝐵𝐶. And we’re trying to find in slope–intercept form the equation of the line that passes through point 𝐴. Slope–intercept form is the form 𝑦 equals 𝑚𝑥 plus 𝑏. That means we need the slope of this line and the 𝑦-intercept. But since we don’t know two points along this line, we’ll have to find the slope a different way.

We remember perpendicular lines have negative reciprocal slopes. And since we do know two points along the line 𝐵𝐶, we can find the slope of line 𝐵𝐶. And the slope along the line that includes point 𝐴 will be equal to negative one over the slope of the line from 𝐵𝐶. This is just a mathematical way to say that these two values will be negative reciprocals of one another. This means our first job is to find the slope of line 𝐵𝐶. If we know two points along the line, we can find their slope by taking 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.

We’ll let 𝐵 be 𝑥 one, 𝑦 one and 𝐶 be 𝑥 two, 𝑦 two. And the slope of line 𝐵𝐶 will be equal to 10 minus negative nine over negative eight minus eight. 10 minus negative nine is 19. And negative eight minus eight equals negative 16. We can say that the slope of line 𝐵𝐶 is 19 over negative 16. But more commonly, we would include the negative sign in the numerator and say the slope of line 𝐵𝐶 equals negative 19 over 16. The slope of the line containing point 𝐴 is the negative reciprocal of this value.

To find the reciprocal of a fraction, we flip it. The reciprocal of negative 19 over 16 is 16 over negative 19. But we have to be careful here because we need the negative reciprocal. And that means negative 16 over negative 19 simplifies to 16 over 19. The slope of the line passing through point 𝐴 is then 16 over 19. At this point, we have the slope of the line passing through point 𝐴. And we have one point that falls along that line.

To find the 𝑦-intercept form of this equation, we could then use the point–slope formula, which says 𝑦 minus 𝑦 one equals 𝑚 times 𝑥 minus 𝑥 one, where 𝑥 one, 𝑦 one is a point along the line. Point 𝐴 is 𝑥 one, 𝑦 one. And so, we have 𝑦 minus negative seven equals 16 over 19 times 𝑥 minus 13. Minus negative seven is plus seven. We distribute that 16 over 19 times 𝑥. And 16 over 19 times negative 13 equals negative 208 over 19.

Since we want the equation in slope–intercept form, we need to get 𝑦 by itself by subtracting seven from both sides. Negative 208 over 19 minus seven is negative 341 over 19. This line 𝑦 equals 16 over 19𝑥 minus 341 over 19 is perpendicular to the line 𝐵𝐶 and passes through point 𝐴.

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