What is an equation of the line that passes through the points left parenthesis, 1, comma, minus, 5, right parenthesis(1,−5) and left parenthesis, 4, comma, 4, right parenthesis(4,4)?

To find the equation of a line that passes through two given points \((1, -5)\) and \((4, 4)\), we can use the point-slope form of a line equation.

### Step 1: Find the slope
The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the given points \((x_1, y_1) = (1, -5)\) and \((x_2, y_2) = (4, 4)\):

\[
m = \frac{4 - (-5)}{4 - 1} = \frac{4 + 5}{4 - 1} = \frac{9}{3} = 3
\]

So, the slope \( m \) is 3.

### Step 2: Use the point-slope form
The point-slope form of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

Using the slope \( m = 3 \) and the point \((1, -5)\), we substitute into the point-slope equation:

\[
y - (-5) = 3(x - 1)
\]

Simplifying:

\[
y + 5 = 3(x - 1)
\]

### Step 3: Simplify to slope-intercept form
Now, distribute and simplify:

\[
y + 5 = 3x - 3
\]

Subtract 5 from both sides:

\[
y = 3x - 3 - 5
\]

\[
y = 3x - 8
\]

### Final Answer:
The equation of the line is:

\[
y = 3x - 8
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the equation of a line passing through the points \((2, 3)\) and \((5, -6)\)?
2. How do you find the slope of a line parallel to \(y = 2x + 3\)?
3. What is the equation of a line with slope \(4\) passing through the point \((2, 1)\)?
4. How do you determine if two lines are perpendicular using their slopes?
5. What is the slope-intercept form of a vertical line passing through \(x = 2\)?

**Tip**: Remember that for vertical lines, the slope is undefined, and the equation is always of the form \(x = c\).

What is an equation of the line that passes through the points (1,−5) and (4,4)?

Learn how to find the equation of a line passing through the points (1, -5) and (4, 4) using the slope formula and point-slope form. This problem demonstrates key concepts in coordinate geometry and is suitable for students in Grades 8-10.

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