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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!

Learn how to write the equation of a linear function passing through the points (2, -4) and (4, 7) using the slope-intercept form. This problem covers calculating slope, finding the y-intercept, and writing the equation of a line. Suitable for students in Grades 7-9.

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