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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=123 x-4 y=12 that contains the point (3,−4)(3,-4) Write the equation of Line 1 in slope-intercept form. □\square Find the slope of Line 1. □\square Find the slope of Line 2. □\square Find the equation of Line 2 in point-slope form using the point (3,−4)(3,-4). □\square Find the equation of Line 2 in the form Ax+By=CA x+B y=C. □\square Need Help? Read It

Studdy Solution

STEP 1

1. Line 1 is given in standard form: 3x−4y=123x - 4y = 12.
2. We need to convert Line 1 to slope-intercept form to find its slope.
3. Line 2 is perpendicular to Line 1 and passes through the point (3,−4)(3, -4).
4. The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
5. We will use the point-slope form to find the equation of Line 2 and then convert it to standard form.

STEP 2

1. Convert the equation of Line 1 to slope-intercept form.
2. Determine the slope of Line 1.
3. Calculate the slope of Line 2.
4. Write the equation of Line 2 in point-slope form.
5. Convert the equation of Line 2 to standard form.

STEP 3

Convert the equation of Line 1, 3x−4y=123x - 4y = 12, to slope-intercept form y=mx+by = mx + b.
To do this, 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

Identify the slope of Line 1 from the slope-intercept form y=34x−3y = \frac{3}{4}x - 3.
The slope mm is 34\frac{3}{4}.

STEP 5

Find the slope of Line 2, which is perpendicular to Line 1.
The slope of Line 2 is the negative reciprocal of 34\frac{3}{4}.
m2=−43 m_2 = -\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 equation of Line 2 from point-slope form to standard form Ax+By=CAx + By = C.
Start with:
y+4=−43(x−3) y + 4 = -\frac{4}{3}(x - 3)
Distribute the slope:
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 standard form:
4x+3y=0 4x + 3y = 0

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