Linear Function y=mx+b: Using slope for geometric problems

To solve this problem, we'll follow these steps:

• Step 1: Identify the coordinates of the points.

• Step 2: Apply the slope formula.

• Step 3: Perform the arithmetic to calculate the slope.

Let's work through each step:

Step 1: Identify the given points:
Point $A$ is $(0, 7)$ and Point $B$ is $(8, -3)$.

Step 2: Use the slope formula, which is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of points $A$ and $B$:
Here, $(x_1, y_1) = (0, 7)$ and $(x_2, y_2) = (8, -3)$.

Step 3: Calculate the slope:
$m = \frac{-3 - 7}{8 - 0} \\ = \frac{-10}{8} \\ = -\frac{5}{4}$

Therefore, the slope of the graph is $-\frac{5}{4}$.

To find the slope of the graph of the linear function passing through points $A(2,10)$ and $B(-5,-4)$, we use the slope formula:

The slope formula is given by:

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

Substitute $(x_1, y_1) = (2, 10)$ and $(x_2, y_2) = (-5, -4)$:

$m = \frac{-4 - 10}{-5 - 2}$

Calculate the differences:

$y_2 - y_1 = -4 - 10 = -14$

$x_2 - x_1 = -5 - 2 = -7$

Substitute these into the slope formula:

$m = \frac{-14}{-7}$

Simplify:

$m = \frac{-14}{-7} = 2$

Therefore, the slope of the graph is $2$.

To determine the slope of the line passing through points $A(-3, 2)$ and $B(3, 2)$, we will use the slope formula:

The slope $m$ is calculated as:

Substituting the values from points $A(-3, 2)$ and $B(3, 2)$, we get:

The calculation shows that the difference in $y$-coordinates is zero, hence dividing by any non-zero number will result in a slope of zero. This indicates a horizontal line on the graph.

Therefore, the slope of the line is $0$.

To solve this problem, we need to calculate the slope of the line passing through the points $A(0, -10)$ and $B(4, 1)$.

The formula for the slope $m$ of a line that passes through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

Given the points $A(0, -10)$ and $B(4, 1)$, we identify:

• $x_1 = 0, y_1 = -10$
• $x_2 = 4, y_2 = 1$

Substituting these values into the slope formula, we have:

This simplifies to:

The fraction $\frac{11}{4}$ can be converted to a mixed number:

Therefore, the slope of the graph is $\bm{2\frac{3}{4}}$.

To find the slope ($m$) of the line passing through the points $A(0,7)$ and $B(-4,-9)$, we apply the slope formula:

First, assign the coordinates to the two points:

• $(x_1, y_1) = (0, 7)$
• $(x_2, y_2) = (-4, -9)$

Next, substitute these into the slope formula:

Simplify the expression:

The negative signs in the numerator and denominator cancel out:

Finally, divide to find the slope:

Therefore, the slope of the line passing through points $A(0,7)$ and $B(-4,-9)$ is $4$.

To find the slope of the linear function passing through the points $A(1,7)$ and $D(8,2)$, we will use the formula for the slope between two points:

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

Let us assign the coordinates $(x_1, y_1) = (1, 7)$ and $(x_2, y_2) = (8, 2)$.

Substitute these values into the slope formula:

$m = \frac{2 - 7}{8 - 1}$

Calculate the differences in the numerator and the denominator:

$m = \frac{-5}{7}$

Therefore, the slope of the line passing through points $A(1,7)$ and $D(8,2)$ is $-\frac{5}{7}$.

In conclusion, the correct answer is $-\frac{5}{7}$.