Select the equation of a line that passes through (10,7) and is perpendicular to the graph of 5x+2y=6
.

	
y=−2.5x+3

	
y=0.4x+7.2

	
y=0.4x+3

	
y=−2.5x+32

To solve this, we need to determine the equation of the line that passes through the point \( (10, 7) \) and is perpendicular to the line given by the equation \( 5x + 2y = 6 \).

### Step 1: Find the slope of the given line
To find the slope, we'll first rewrite the equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope.

Start with the given equation:
\[
5x + 2y = 6
\]

Solve for \( y \):
\[
2y = -5x + 6
\]
\[
y = -\frac{5}{2}x + 3
\]

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

### Step 2: Find the slope of the perpendicular line
For two lines to be perpendicular, their slopes must multiply to give \( -1 \). The slope of the perpendicular line, \( m_2 \), can be found by taking the negative reciprocal of \( m_1 \).

\[
m_2 = \frac{2}{5}
\]

### Step 3: Use the point-slope form to find the equation of the perpendicular line
Now we use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
where \( m = \frac{2}{5} \) is the slope, and \( (x_1, y_1) = (10, 7) \) is the point the line passes through.

Substitute the values:
\[
y - 7 = \frac{2}{5}(x - 10)
\]
Distribute the slope:
\[
y - 7 = \frac{2}{5}x - 4
\]
Add 7 to both sides:
\[
y = \frac{2}{5}x + 3
\]

Thus, the equation of the line is \( y = 0.4x + 3 \).

### Step 4: Select the correct option
The correct equation is \( y = 0.4x + 3 \).

### Final Answer:
\[
\boxed{y = 0.4x + 3}
\]

Would you like more details on any step or have any questions?

Here are 5 related questions you might find interesting:
1. How do you find the slope of a line given two points?
2. What is the general method for finding the equation of a line parallel to a given line?
3. How do you find the point of intersection of two lines?
4. What is the relationship between the slopes of parallel and perpendicular lines?
5. How do you convert a line equation from standard form to slope-intercept form?

**Tip**: When dealing with perpendicular lines, always remember that their slopes are negative reciprocals of each other!

Select the equation of a line that passes through (10,7) and is perpendicular to the graph of 5x+2y=6.

Learn how to find the equation of a line that passes through the point (10, 7) and is perpendicular to the line 5x + 2y = 6. This problem covers key algebraic concepts like slope and perpendicular lines. Suitable for students in Grades 9-11.

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