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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.
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So, here’s what we’re thinking.
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We have points 𝐵 and 𝐶, which form a line.
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Point 𝐴 does not fall on this line.
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But point 𝐴 falls on a line perpendicular to the line 𝐵𝐶.
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And we’re trying to find in slope–intercept form the equation of the line that passes through point 𝐴.
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Slope–intercept form is the form 𝑦 equals 𝑚𝑥 plus 𝑏.
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That means we need the slope of this line and the 𝑦-intercept.
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But since we don’t know two points along this line, we’ll have to find the slope a different way.
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We remember perpendicular lines have negative reciprocal slopes.
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And since we do know two points along the line 𝐵𝐶, we can find the slope of line 𝐵𝐶.
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And the slope along the line that includes point 𝐴 will be equal to negative one over the slope of the line from 𝐵𝐶.
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This is just a mathematical way to say that these two values will be negative reciprocals of one another.
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This means our first job is to find the slope of line 𝐵𝐶.
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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.
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We’ll let 𝐵 be 𝑥 one, 𝑦 one and 𝐶 be 𝑥 two, 𝑦 two.
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And the slope of line 𝐵𝐶 will be equal to 10 minus negative nine over negative eight minus eight.
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10 minus negative nine is 19.
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And negative eight minus eight equals negative 16.
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We can say that the slope of line 𝐵𝐶 is 19 over negative 16.
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But more commonly, we would include the negative sign in the numerator and say the slope of line 𝐵𝐶 equals negative 19 over 16.
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The slope of the line containing point 𝐴 is the negative reciprocal of this value.
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To find the reciprocal of a fraction, we flip it.
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The reciprocal of negative 19 over 16 is 16 over negative 19.
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But we have to be careful here because we need the negative reciprocal.
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And that means negative 16 over negative 19 simplifies to 16 over 19.
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The slope of the line passing through point 𝐴 is then 16 over 19.
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At this point, we have the slope of the line passing through point 𝐴.
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And we have one point that falls along that line.
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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.
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Point 𝐴 is 𝑥 one, 𝑦 one.
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And so, we have 𝑦 minus negative seven equals 16 over 19 times 𝑥 minus 13.
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Minus negative seven is plus seven.
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We distribute that 16 over 19 times 𝑥.
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And 16 over 19 times negative 13 equals negative 208 over 19.
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Since we want the equation in slope–intercept form, we need to get 𝑦 by itself by subtracting seven from both sides.
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Negative 208 over 19 minus seven is negative 341 over 19.
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This line 𝑦 equals 16 over 19𝑥 minus 341 over 19 is perpendicular to the line 𝐵𝐶 and passes through point 𝐴.