The problem asks to write an equation for the linear function that passes through the points $(2, -4)$ and $(4, 7)$. The equation should be given in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Let's go through the solution step by step.

Step 1: Find the Slope (m)

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Using the points $(2, -4)$ and $(4, 7)$:

$m = \frac{7 - (-4)}{4 - 2} = \frac{7 + 4}{2} = \frac{11}{2}$

So, the slope $m$ is $\frac{11}{2}$.

Step 2: Use the Slope-Intercept Form (y = mx + b)

Now that we know the slope, we can use the slope-intercept form $y = mx + b$. We need to find the y-intercept $b$.

Step 3: Find the y-Intercept (b)

We can use one of the given points to find $b$. Let's use the point $(2, -4)$.

Substitute $x = 2$, $y = -4$, and $m = \frac{11}{2}$ into the equation $y = mx + b$:

$-4 = \frac{11}{2}(2) + b$

Simplifying:

$-4 = 11 + b$

Now, solve for $b$:

$b = -4 - 11 = -15$

Final Equation

Now that we have $m = \frac{11}{2}$ and $b = -15$, the equation of the line in slope-intercept form is:

$y = \frac{11}{2}x - 15$

This is the required linear equation.

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Do you want further details or have any questions?

Here are 5 related questions for practice:

1. Find the equation of a line passing through $(1, 3)$ and $(3, 9)$ in slope-intercept form.
2. Determine the slope of a line that passes through $(-2, 5)$ and $(4, -1)$.
3. Write the equation of the line with slope $7$ and passing through the point $(0, -3)$.
4. Convert the equation $2x + 3y = 12$ to slope-intercept form.
5. Find the y-intercept of the line passing through the points $(6, 8)$ and $(0, 2)$.

Tip: Always substitute a known point into the equation of the line to verify that the equation is correct!