Find an equation for the line with the given property.
(a)
Passes through the points 
(6, 5) and (1, −5)

To find the equation of the line passing through the points \((6, 5)\) and \((1, -5)\), follow these steps:

### 1. **Calculate the Slope (\(m\)):**
The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the given points \((6, 5)\) and \((1, -5)\):
\[
m = \frac{-5 - 5}{1 - 6} = \frac{-10}{-5} = 2
\]

### 2. **Use the Point-Slope Form to Find the Equation:**
The point-slope form of a line's equation is:
\[
y - y_1 = m(x - x_1)
\]

Using the slope \(m = 2\) and one of the points, say \((6, 5)\):
\[
y - 5 = 2(x - 6)
\]

### 3. **Simplify to the Slope-Intercept Form:**
Expand and simplify this equation to get it in the slope-intercept form \(y = mx + b\):
\[
y - 5 = 2x - 12
\]
\[
y = 2x - 12 + 5
\]
\[
y = 2x - 7
\]

So, the equation of the line passing through the points \((6, 5)\) and \((1, -5)\) is:
\[
\boxed{y = 2x - 7}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you find the equation of a line given a slope and one point?
2. How can you determine if two lines are parallel or perpendicular?
3. What is the point-slope form of a line, and how is it derived?
4. How can you find the x-intercept and y-intercept of a line from its equation?
5. How do you convert a linear equation from standard form to slope-intercept form?

**Tip:** Always verify your calculations by substituting the coordinates of the points back into the final equation to ensure they satisfy it.

Find an equation for the line with the given property. (a) Passes through the points (6, 5) and (1, −5).

Find the equation of the line passing through the points (6, 5) and (1, −5). This problem involves calculating the slope, using the point-slope form, and converting to slope-intercept form. Detailed solution with step-by-step guidance, suitable for Grades 8-10.

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