To find the equation of the linear function that passes through the points $(-1, -1)$ and $(10, 4)$, we will use the point-slope form of the linear equation:

$y - y_1 = m(x - x_1)$

Where:

• $m$ is the slope of the line,
• $(x_1, y_1)$ is one of the points on the line.

Step 1: Find the slope $m$

The slope of a line is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(-1, -1)$ and $(10, 4)$:

$m = \frac{4 - (-1)}{10 - (-1)} = \frac{4 + 1}{10 + 1} = \frac{5}{11}$

Step 2: Use the point-slope form

Now that we know the slope $m = \frac{5}{11}$, we can use one of the points (let's use $(-1, -1)$) to find the equation of the line:

$y - (-1) = \frac{5}{11}(x - (-1))$

Simplifying:

$y + 1 = \frac{5}{11}(x + 1)$

Now, distribute the slope:

$y + 1 = \frac{5}{11}x + \frac{5}{11}$

Step 3: Solve for $y$

$y = \frac{5}{11}x + \frac{5}{11} - 1$

Simplify the constant term:

$y = \frac{5}{11}x + \frac{5}{11} - \frac{11}{11}$

$y = \frac{5}{11}x - \frac{6}{11}$

Final Answer:

The equation of the line passing through the points $(-1, -1)$ and $(10, 4)$ is:

$y = \frac{5}{11}x - \frac{6}{11}$

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Would you like further details or clarification? Here are some related questions for practice:

1. How would the equation change if the points were $(2, 3)$ and $(5, 8)$?
2. What does the slope of the line represent in real-world terms?
3. How do you convert this linear equation to slope-intercept form?
4. Can you find the equation of a line given the slope and a point on the line?
5. How would you graph this equation on a coordinate plane?

Tip: When working with linear equations, always double-check the calculations for the slope and intercepts to avoid small mistakes.