To find the slope-intercept form of the equation of a line, $y = mx + b$, we need to determine the slope $m$ and the y-intercept $b$.

Step 1: Find the slope $m$

The slope $m$ is given by the formula:

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

Using the points $(1, 1)$ and $(7, -\frac{3}{4})$, assign:

• $x_1 = 1$, $y_1 = 1$
• $x_2 = 7$, $y_2 = -\frac{3}{4}$

Substitute these values into the slope formula:

$m = \frac{-\frac{3}{4} - 1}{7 - 1} = \frac{-\frac{3}{4} - \frac{4}{4}}{6} = \frac{-\frac{7}{4}}{6} = \frac{-7}{24}$

So, the slope is $m = -\frac{7}{24}$.

Step 2: Use point-slope form to find the equation

Now that we know the slope $m = -\frac{7}{24}$, use the point-slope form of the line equation:

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

Substitute $m = -\frac{7}{24}$, $x_1 = 1$, and $y_1 = 1$:

$y - 1 = -\frac{7}{24}(x - 1)$

Step 3: Simplify to slope-intercept form

Distribute the slope on the right side:

$y - 1 = -\frac{7}{24}x + \frac{7}{24}$

Add 1 to both sides to solve for $y$:

$y = -\frac{7}{24}x + \frac{7}{24} + 1$

Convert 1 to a fraction with a denominator of 24:

$y = -\frac{7}{24}x + \frac{7}{24} + \frac{24}{24}$

Simplify the constant term:

$y = -\frac{7}{24}x + \frac{31}{24}$

Final Equation

The slope-intercept form of the line is:

$y = -\frac{7}{24}x + \frac{31}{24}$

Would you like further details or clarifications?

Here are 5 related questions:

1. How do you find the slope if the points involve negative coordinates?
2. How does the slope-intercept form compare to the standard form of a line?
3. What is the significance of the slope in real-world applications?
4. How would you graph the line given this equation?
5. How does the point-slope form relate to the slope-intercept form?

Tip: The slope of a line represents how steep the line is; a negative slope indicates that the line is decreasing as you move from left to right.