To find the linear function (or equation of the line) passing through the points $(1, -1)$ and $(4, 3)$, we will use the slope-intercept form of a linear equation, which is:

$y = mx + b$

where:

• $m$ is the slope of the line
• $b$ is the y-intercept

Step 1: Find the slope $m$

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

Substitute the coordinates $(1, -1)$ and $(4, 3)$ into the formula:

$m = \frac{3 - (-1)}{4 - 1} = \frac{3 + 1}{3} = \frac{4}{3}$

So, the slope $m = \frac{4}{3}$.

Step 2: Use the slope and one point to find the y-intercept $b$

Now, we can use the slope-intercept form $y = mx + b$ to find $b$. We already know $m = \frac{4}{3}$, and we can use one of the points (let's use $(1, -1)$) to solve for $b$.

Substitute $x = 1$ and $y = -1$ into the equation $y = mx + b$:

$-1 = \frac{4}{3}(1) + b$

Now solve for $b$:

$-1 = \frac{4}{3} + b$

Subtract $\frac{4}{3}$ from both sides:

$b = -1 - \frac{4}{3} = \frac{-3}{3} - \frac{4}{3} = \frac{-7}{3}$

So, $b = \frac{-7}{3}$.

Step 3: Write the equation of the line

Now that we know the slope and the y-intercept, we can write the equation of the line:

$y = \frac{4}{3}x - \frac{7}{3}$

This is the linear function passing through the points $(1, -1)$ and $(4, 3)$.

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Would you like more details on the steps or have any other questions about linear functions?

Here are 5 related questions to explore:

1. How can we graph this linear equation?
2. What is the meaning of the slope and y-intercept in the context of the line?
3. How do we find the equation of a line given a point and the slope?
4. What if the points had been vertical or horizontal—how would the equation change?
5. How can you determine if two lines are parallel or perpendicular based on their slopes?

Tip: To check your linear function, you can substitute the coordinates of the second point into the equation and see if it satisfies the equation.