Find the equation of a line perpendicular toy, plus, 5, equals, start fraction, 3, divided by, 2, end fraction, xy+5= 
2
3
​
 xthat passes through the point left parenthesis, 3, comma, 4, right parenthesis(3,4).
Answer
Multiple Choice Answers
y, equals, start fraction, 3, divided by, 2, end fraction, x, plus, 5y= 
2
3
​
 x+5
y, equals, start fraction, 2, divided by, 3, end fraction, x, plus, 6y= 
3
2
​
 x+6
y, equals, minus, start fraction, 3, divided by, 2, end fraction, x, plus, 5y=− 
2
3
​
 x+5
y, equals, minus, start fraction, 2, divided by, 3, end fraction, x, plus, 6y=− 
3
2
​
 x+6

We are asked to find the equation of a line perpendicular to the given line and passing through the point \((3,4)\).

### Step 1: Identify the slope of the given line

The equation provided is:
\[
y + 5 = \frac{3}{2}x
\]
Rearranging it into slope-intercept form \(y = mx + b\), we subtract 5 from both sides:
\[
y = \frac{3}{2}x - 5
\]
This gives us the slope of the given line, \(m = \frac{3}{2}\).

### Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, if the slope of the original line is \(\frac{3}{2}\), the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{2}{3}
\]

### Step 3: Use the point-slope form

Now, we know that the perpendicular line has a slope of \(-\frac{2}{3}\) and passes through the point \((3,4)\). We use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting the point \((3, 4)\) and the slope \(-\frac{2}{3}\):
\[
y - 4 = -\frac{2}{3}(x - 3)
\]

### Step 4: Simplify the equation

Distribute the slope on the right-hand side:
\[
y - 4 = -\frac{2}{3}x + 2
\]
Now, add 4 to both sides:
\[
y = -\frac{2}{3}x + 6
\]

### Step 5: Match the answer

The equation of the perpendicular line is:
\[
y = -\frac{2}{3}x + 6
\]

This matches the option:
\[
\boxed{y = -\frac{2}{3}x + 6}
\]

### Would you like more details or have any questions?

Here are five related questions for practice:
1. What is the equation of a line with slope \(\frac{3}{4}\) passing through \((2, -1)\)?
2. How do you find the equation of a line parallel to \(y = 4x + 3\) passing through \((1, 2)\)?
3. What is the slope of a line perpendicular to \(y = -\frac{5}{3}x + 7\)?
4. How do you convert the point-slope form of a line to slope-intercept form?
5. Find the slope of a line that passes through points \((0,0)\) and \((5,10)\).

Tip: The slope of perpendicular lines is always the negative reciprocal of the original slope.

Find the equation of a line perpendicular to y + 5 = (3/2)x that passes through the point (3, 4).

Learn how to find the equation of a line perpendicular to y + 5 = (3/2)x that passes through the point (3, 4). This solution explains the concept of perpendicular slopes and uses point-slope form to derive the final equation. Suitable for high school students.

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