Math  /  Geometry

QuestionSelect all the lines that are perpendicular to 3xy=103 x-y=10. A. y=3x+5y=3 x+5 B. y=13x+17y=-\frac{1}{3} x+17 C. x+3y=27x+3 y=27 D. y2=13(3x+36)y-2=\frac{1}{3}(3 x+36) E. y=13x+2y=\frac{1}{3} x+2

Studdy Solution

STEP 1

What is this asking? Which of these equations represent lines that meet 3xy=103x - y = 10 at a right angle? Watch out! Don't forget, perpendicular lines have slopes that are *negative reciprocals* of each other!

STEP 2

1. Find the Slope
2. Check Each Line

STEP 3

We're given 3xy=103x - y = 10.
Let's **rewrite this equation** in slope-intercept form (y=mx+by = mx + b), where mm is the **slope** and bb is the **y-intercept**.
To do this, we want to isolate yy.
First, we'll subtract 3x3x from both sides to get y=3x+10-y = -3x + 10.

STEP 4

Now, we'll multiply both sides by 1-1 to make the **coefficient** of yy positive.
This gives us y=3x10y = 3x - 10.

STEP 5

Now we can see that the **slope** of our given line is m=3m = \mathbf{3}.

STEP 6

The equation y=3x+5y = 3x + 5 has a slope of 33.
Since 33 is not the negative reciprocal of 33, this line is *not* perpendicular to our given line.

STEP 7

y=13x+17y = -\frac{1}{3}x + 17 has a slope of 13-\frac{1}{3}.
This *is* the negative reciprocal of 33 (because 313=13 \cdot -\frac{1}{3} = -1), so line B *is* perpendicular!

STEP 8

We need to rewrite x+3y=27x + 3y = 27 in slope-intercept form.
Subtracting xx from both sides gives 3y=x+273y = -x + 27.
Dividing both sides by 33 gives y=13x+9y = -\frac{1}{3}x + 9.
The slope is 13-\frac{1}{3}, which is the negative reciprocal of 33, so line C *is* perpendicular!

STEP 9

Let's rewrite y2=13(3x+36)y - 2 = \frac{1}{3}(3x + 36).
Distributing the 13\frac{1}{3} gives us y2=x+12y - 2 = x + 12.
Adding 22 to both sides gives y=x+14y = x + 14.
The slope is 11, which is not the negative reciprocal of 33, so this line is *not* perpendicular.

STEP 10

y=13x+2y = \frac{1}{3}x + 2 has a slope of 13\frac{1}{3}.
This is not the negative reciprocal of 33, so this line is *not* perpendicular.

STEP 11

Lines B and C are perpendicular to the given line.

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