1.Find the equation of the line passing through the point 
(
−
6
,
4
)
(−6,4) that is perpendicular to the line 
5
x
+
3
y
=
5
5x+3y=5.

y
=
3
5
x
−
14
5
y= 
5
3
​
 x− 
5
14
​
 

y
=
3
5
x
+
38
5
y= 
5
3
​
 x+ 
5
38
​
 

y
=
−
5
3
x
−
4
3
y=− 
3
5
​
 x− 
3
4
​
 

y
=
−
5
3
x
+
4
3
y=− 
3
5
​
 x+ 
3
4
​
 

y
=
3
5
x
−
38
5
y= 
5
3
​
 x− 
5
38
​
]

To find the equation of a line that passes through the point \((-6, 4)\) and is perpendicular to the line \(5x + 3y = 5\), follow these steps:

### Step 1: Find the Slope of the Given Line
The equation of the line is \(5x + 3y = 5\). We start by rewriting it in slope-intercept form \(y = mx + b\) to identify its slope.

1. **Rewrite the equation**:
   \[
   5x + 3y = 5
   \]
   Subtract \(5x\) from both sides:
   \[
   3y = -5x + 5
   \]
2. **Solve for \(y\)** by dividing every term by 3:
   \[
   y = -\frac{5}{3}x + \frac{5}{3}
   \]

So, the slope \(m\) of this line is \(-\frac{5}{3}\).

### Step 2: Find the Perpendicular Slope
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the perpendicular slope \(m_{\perp}\) is:
\[
m_{\perp} = \frac{3}{5}
\]

### Step 3: Use the Point-Slope Formula
Now we have a point \((-6, 4)\) and a slope \(\frac{3}{5}\). Use the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \(x_1 = -6\), \(y_1 = 4\), and \(m = \frac{3}{5}\):
\[
y - 4 = \frac{3}{5}(x + 6)
\]

### Step 4: Simplify the Equation
Distribute \(\frac{3}{5}\):
\[
y - 4 = \frac{3}{5}x + \frac{18}{5}
\]
Add 4 to both sides to isolate \(y\):
\[
y = \frac{3}{5}x + \frac{18}{5} + \frac{20}{5}
\]
Combine the constants:
\[
y = \frac{3}{5}x + \frac{38}{5}
\]

### Final Answer
The equation of the line is:
\[
y = \frac{3}{5}x + \frac{38}{5}
\]

So the correct answer is:
\[
y = \frac{3}{5}x + \frac{38}{5}
\]

Let me know if you would like further details or have questions.

---

### Related Practice Questions

1. What would be the equation of a line parallel to \(5x + 3y = 5\) that passes through \((4, -1)\)?
2. If a line passes through \((-6, 4)\) and has a slope of \(\frac{3}{5}\), what is its equation in standard form?
3. How do you determine if two lines are perpendicular just from their slopes?
4. What is the distance between the point \((-6, 4)\) and the line \(5x + 3y = 5\)?
5. How would the equation change if the line had to pass through a different point, like \((0, 0)\)?

---

### Tip
When finding perpendicular lines, remember that the slopes multiply to \(-1\), as they are negative reciprocals.

Find the equation of the line passing through the point (-6, 4) that is perpendicular to the line 5x + 3y = 5.

Learn how to find the equation of a line that passes through the point (-6, 4) and is perpendicular to the line 5x + 3y = 5. This problem involves key concepts in algebra, including the point-slope formula and properties of perpendicular lines. Suitable for students in Grades 8-10.

Math Problem Statement

1.Find the equation of the line passing through the point ( − 6 , 4 ) (−6,4) that is perpendicular to the line 5 x + 3 y

y

y

y

y

y

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