Write the equation of a line that is perpendicular to the line 5x+2y=1 and goes through the point (0,3).

To write the equation of a line perpendicular to the line \( 5x + 2y = 1 \) and passing through the point \( (0, 3) \), follow these steps:

### Step 1: Find the slope of the given line
First, we rewrite the equation \( 5x + 2y = 1 \) in slope-intercept form (\( y = mx + b \)), where \( m \) is the slope.

\[
5x + 2y = 1
\]
Solve for \( y \):
\[
2y = -5x + 1
\]
\[
y = -\frac{5}{2}x + \frac{1}{2}
\]

The slope of the given line is \( m_1 = -\frac{5}{2} \).

### Step 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. So, the slope of the perpendicular line, \( m_2 \), is:
\[
m_2 = \frac{2}{5}
\]

### Step 3: Write the equation of the new line
Now that we know the slope \( m_2 = \frac{2}{5} \) and the line passes through the point \( (0, 3) \), we use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m_2 = \frac{2}{5} \), \( x_1 = 0 \), and \( y_1 = 3 \):
\[
y - 3 = \frac{2}{5}(x - 0)
\]
\[
y - 3 = \frac{2}{5}x
\]
\[
y = \frac{2}{5}x + 3
\]

Thus, the equation of the line is:
\[
y = \frac{2}{5}x + 3
\]

Would you like further clarification or more details?

### 5 Related Questions:
1. How do you find the slope of a line from its general form equation?
2. What is the negative reciprocal of a slope, and why does it give the slope of a perpendicular line?
3. How can you convert a line equation from point-slope form to slope-intercept form?
4. How do you find the equation of a line parallel to a given line?
5. What are the properties of perpendicular lines in terms of their slopes?

### Tip:
Remember, for two lines to be perpendicular, the product of their slopes must always be \( -1 \).

Write the equation of a line that is perpendicular to the line 5x + 2y = 1 and goes through the point (0, 3).

Learn how to find the equation of a line that is perpendicular to 5x + 2y = 1 and passes through the point (0, 3). This problem covers key algebraic concepts such as slopes, perpendicular lines, and the slope-intercept form. Suitable for students in Grades 8-10.

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