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Math  /  Algebra

QuestionConsider the following lines. Line 1: 3x−4y=123 x-4 y=12 Line 2: a line perpendicular to 3x−4y=423 x-4 y=42 that contains the point (3,−4)(3,-4) Write the equation of Line 1 in slope-intercept form. y=34x−3y=\frac{3}{4} x-3
Find the slope of Line 1. 3/43 / 4
Find the slope of Line 2. −4/3-4 / 3
Find the equation of Line 2 in point-slope form using the point (3,−4)(3,-4). y+4=−43(x−3)y+4=-\frac{4}{3}(x-3)
Find the equation of Line 2 in the form Ax+By=CA x+B y=C. □\square

Studdy Solution

STEP 1

1. Line 1 is given by the equation 3x−4y=123x - 4y = 12.
2. Line 2 is perpendicular to Line 1 and passes through the point (3,−4)(3, -4).
3. We need to find the equation of Line 2 in the form Ax+By=CAx + By = C.

STEP 2

1. Convert the equation of Line 1 to slope-intercept form.
2. Determine the slope of Line 1.
3. Find the slope of Line 2, which is perpendicular to Line 1.
4. Write the equation of Line 2 in point-slope form.
5. Convert the equation of Line 2 to the form Ax+By=CAx + By = C.

STEP 3

Convert the equation of Line 1, 3x−4y=123x - 4y = 12, to slope-intercept form y=mx+by = mx + b.
Rearrange the equation to solve for yy:
3x−4y=12 3x - 4y = 12
Subtract 3x3x from both sides:
−4y=−3x+12 -4y = -3x + 12
Divide every term by −4-4:
y=34x−3 y = \frac{3}{4}x - 3

STEP 4

The slope-intercept form of Line 1 is y=34x−3y = \frac{3}{4}x - 3.
Thus, the slope of Line 1 is:
34 \frac{3}{4}

STEP 5

The slope of Line 2, which is perpendicular to Line 1, is the negative reciprocal of the slope of Line 1.
The slope of Line 1 is 34\frac{3}{4}, so the slope of Line 2 is:
−43 -\frac{4}{3}

STEP 6

Write the equation of Line 2 in point-slope form using the point (3,−4)(3, -4) and the slope −43-\frac{4}{3}.
The point-slope form is:
y−y1=m(x−x1) y - y_1 = m(x - x_1)
Substitute m=−43m = -\frac{4}{3}, x1=3x_1 = 3, and y1=−4y_1 = -4:
y+4=−43(x−3) y + 4 = -\frac{4}{3}(x - 3)

STEP 7

Convert the point-slope form of Line 2, y+4=−43(x−3)y + 4 = -\frac{4}{3}(x - 3), to the form Ax+By=CAx + By = C.
Distribute −43-\frac{4}{3}:
y+4=−43x+4 y + 4 = -\frac{4}{3}x + 4
Subtract 4 from both sides:
y=−43x y = -\frac{4}{3}x
Multiply every term by 3 to eliminate the fraction:
3y=−4x 3y = -4x
Rearrange to get the standard form:
4x+3y=0 4x + 3y = 0
The equation of Line 2 in the form Ax+By=CAx + By = C is:
4x+3y=0 \boxed{4x + 3y = 0}

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