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The problem in the image asks for the equation of a line that passes through the point \((5, 2)\) and is perpendicular to the line given by the equation \(5x + 4y = 4\). The equation should be written in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### Step-by-step solution:

1. **Rewrite the given equation in slope-intercept form**:
   To find the slope of the given line \(5x + 4y = 4\), solve for \(y\).
   \[
   4y = -5x + 4
   \]
   \[
   y = -\frac{5}{4}x + 1
   \]
   So, the slope of this line is \(m = -\frac{5}{4}\).

2. **Find the slope of the perpendicular line**:
   The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Since the slope of the given line is \(-\frac{5}{4}\), the slope of the perpendicular line is:
   \[
   m_{\text{perpendicular}} = \frac{4}{5}
   \]

3. **Use the point-slope form to find the equation of the new line**:
   We are given that the line passes through the point \((5, 2)\) and has a slope of \(\frac{4}{5}\). Using the point-slope form \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point \((5, 2)\), we can write:
   \[
   y - 2 = \frac{4}{5}(x - 5)
   \]
   Simplifying this equation:
   \[
   y - 2 = \frac{4}{5}x - 4
   \]
   \[
   y = \frac{4}{5}x - 4 + 2
   \]
   \[
   y = \frac{4}{5}x - 2
   \]

### Final Answer:
The equation of the line that passes through the point \((5, 2)\) and is perpendicular to the line \(5x + 4y = 4\) is:
\[
y = \frac{4}{5}x - 2
\]

Would you like any further details or explanations on this problem?

Here are five related questions you might consider:
1. How do you find the slope of a line from a general equation like \(Ax + By = C\)?
2. What is the significance of the negative reciprocal when determining perpendicular lines?
3. How can you determine the y-intercept of a line if only the slope and a point are given?
4. How do parallel and perpendicular lines differ in terms of their slopes?
5. Can the point-slope form be used to find the equation of a line in every case?

**Tip**: Always remember that perpendicular lines have slopes that are negative reciprocals of each other. This is crucial when solving geometry problems involving line equations.

The equation of the line that goes through the point (5, 2) and is perpendicular to the line 5x + 4y = 4 can be written in the form y = mx + b.

Find the equation of the line that passes through the point (5, 2) and is perpendicular to 5x + 4y = 4. This problem focuses on linear equations, slopes, and perpendicular lines, suitable for students in Grades 9-12.

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