Graphical Representation: Using additional geometric shapes

To solve this problem, we will calculate the slope of the line passing through the given points:

• Step 1: Identify coordinates: The points are $(5, 0)$ and $(0, -5)$.
• Step 2: Use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the coordinates, we have: $m = \frac{-5 - 0}{0 - 5}$
• Step 3: Simplify the expression: $m = \frac{-5}{-5} = 1$

Thus, the slope of the line is $1$.

Therefore, the solution to the problem is $1$.

To solve this problem, we'll determine the slope of the line that passes through the points $A(0, 10)$ and $B(5, 0)$. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

We identify the coordinates of our points:

• $(x_1, y_1) = (5, 0)$
• $(x_2, y_2) = (0, 10)$.

Substitute these coordinates into the slope formula:

$m = \frac{10 - 0}{0 - 5}$.

Simplifying the equation gives:

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

Thus, the slope of the line is $-2$.

Therefore, the solution to the problem is $m = -2$.

To determine the slopes of the two lines, we will use the slope formula:

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's start with the first line:

Line I:

The points are $(4, 0)$ and $(0, 2)$. Applying the slope formula:

$m_1 = \frac{2 - 0}{0 - 4} = \frac{2}{-4} = -\frac{1}{2}$

Thus, the slope of Line I is $-\frac{1}{2}$.

Now, let's calculate the slope for the second line:

Line II:

The points are $(0, -6)$ and $(4, 0)$. Applying the slope formula:

$m_2 = \frac{0 - (-6)}{4 - 0} = \frac{6}{4} = \frac{3}{2}$

Therefore, the slope of Line II is $\frac{3}{2}$.

In conclusion, the slopes are as follows:

$\frac{3}{2} = II \text{ , } -\frac{1}{2} = I$.

The solution matches choice $\text{Choice 3}$.

The correct answer to the problem is: $\frac{3}{2} = II, -\frac{1}{2} = I$.

To solve this problem, we'll calculate the slope using the slope formula:

• Step 1: Identify the coordinates of the points. We have point $A(0, 0)$ and point $B(5, -2)$.
• Step 2: Apply the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Step 3: Substitute the coordinates: $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (5, -2)$.

Now, let's work through each step:

The slope formula is:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 0}{5 - 0}$.

This simplifies to:
$m = \frac{-2}{5}$.

Therefore, the solution to the problem is $m = -\frac{2}{5}$.

The correct answer choice is $-\frac{2}{5}$.

To find the equation of the line passing through the points $(8,3)$ and $(2,7)$, follow these steps:

• Step 1: Calculate the slope $m$.
• Step 2: Use the slope and one of the points to solve for the y-intercept $b$.
• Step 3: Write the equation of the line in the form $y = mx + b$.

Step 1: Calculate the slope $m$. The slope $m$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{2 - 8} = \frac{4}{-6} = -\frac{2}{3}$.

Step 2: With the slope known, use one point (for example, $(8,3)$) to find $b$ in the slope-intercept form $y = mx + b$. Substitute the values:

$3 = -\frac{2}{3}(8) + b$.

This simplifies to:

$3 = -\frac{16}{3} + b$.

Add $\frac{16}{3}$ to both sides to solve for $b$:

$b = 3 + \frac{16}{3} = \frac{9}{3} + \frac{16}{3} = \frac{25}{3}$.

Step 3: Write the equation using the calculated slope and y-intercept:

$y = -\frac{2}{3}x + \frac{25}{3}$.

To express it as a mixed number, $\frac{25}{3}$ is $8\frac{1}{3}$, so:

$y = -\frac{2}{3}x + 8\frac{1}{3}$.

Thus, the correct equation of the line is $y = -\frac{2}{3}x + 8\frac{1}{3}$, which corresponds to choice 3.

The final answer is: $y = -\frac{2}{3}x + 8\frac{1}{3}$.

Let's derive the equation of the line:

• Step 1: Calculate the Slope
  The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is computed as follows: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the given points $(0, -6)$ and $(4, 0)$: $m = \frac{0 - (-6)}{4 - 0} = \frac{6}{4} = \frac{3}{2}$ Hence, the slope of the line is $\frac{3}{2}$.

• Step 2: Write the Equation Using the Slope-Intercept Form
  The slope-intercept form is: $y = mx + b$ Where $m$ is the slope and $b$ is the y-intercept. Since the line passes through $(0, -6)$, this point is the y-intercept ($b = -6$). Thus, we have: } $y = \frac{3}{2}x - 6$

Therefore, the equation of the line is $y = \frac{3}{2}x - 6$.

The correct choice is option 4.