(8,5); perpendicular to 2x-y=7 write the equation of the line in slope intercept form

To find the equation of a line perpendicular to \( 2x - y = 7 \), and passing through the point \( (8, 5) \), we can follow these steps:

### Step 1: Find the slope of the given line.
The given line is in standard form: 
\[
2x - y = 7
\]
We convert this to slope-intercept form \( y = mx + b \), where \( m \) is the slope:
\[
y = 2x - 7
\]
The slope of the given line is \( m = 2 \).

### 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. So, if the slope of the given line is \( 2 \), the slope of the perpendicular line is:
\[
m_{\text{perpendicular}} = -\frac{1}{2}
\]

### Step 3: Use the point-slope form to find the equation.
We use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = -\frac{1}{2} \), \( (x_1, y_1) = (8, 5) \):
\[
y - 5 = -\frac{1}{2}(x - 8)
\]

### Step 4: Simplify to slope-intercept form.
Distribute and simplify the equation:
\[
y - 5 = -\frac{1}{2}x + 4
\]
\[
y = -\frac{1}{2}x + 4 + 5
\]
\[
y = -\frac{1}{2}x + 9
\]

### Final Answer:
The equation of the line in slope-intercept form is:
\[
y = -\frac{1}{2}x + 9
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the equation of a line parallel to a given line?
2. How do you convert a line equation from slope-intercept form to standard form?
3. How can you check if two lines are perpendicular?
4. What is the general form of a line equation, and how is it related to slope-intercept form?
5. How do you graph lines using the slope and y-intercept?

**Tip:** Always remember, for two lines to be perpendicular, their slopes must multiply to \(-1\).

Find the equation of a line perpendicular to 2x - y = 7 and passing through the point (8, 5). Write the equation in slope-intercept form.

Learn how to find the equation of a line perpendicular to 2x - y = 7 and passing through the point (8, 5). This problem explores key algebraic concepts such as slope-intercept form, point-slope form, and the relationship between perpendicular lines. Detailed step-by-step solution provided, suitable for Grades 8-10.

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