Equation of a Straight Line: Using the angle formed by the line with the axes

To find which line forms a greater angle with the x-axis, we use the formula $\theta = \tan^{-1}(m)$. This gives us the angles formed by lines with slopes $m_1 = -3$ and $m_2 = -6$.

We first calculate the inverse tangent for both slopes:

• For $m_1 = -3$, the angle is $\theta_1 = \tan^{-1}(-3)$.
• For $m_2 = -6$, the angle is $\theta_2 = \tan^{-1}(-6)$.

The greater the absolute value of the negative slope, the closer the line is to the vertical, thus forming a greater angle with the x-axis.

Comparing $|-3|$ and $|-6|$, we see that $-6$ has a greater absolute value, indicating a steeper angle. Hence, although $-6$ is steeper, it forms a smaller angle with the x-axis because we're considering angles formed in the positive x-direction (i.e., above the x-axis).

Therefore, the line whose slope is $-3$ forms the greater angle with the x-axis.

The line whose slope is -3 forms the greater angle.

To solve for which line forms a larger angle with the x-axis, we'll proceed by calculating the arctangent of each slope:

• First, calculate the angle for the line with slope $m_1 = 2$:

$\theta_1 = \tan^{-1}(2)$

• Next, calculate the angle for the line with slope $m_2 = \frac{1}{2}$:

$\theta_2 = \tan^{-1}\left(\frac{1}{2}\right)$

Comparing these two results:

• The function $\tan^{-1}(x)$ is increasing, meaning as the value of $x$ increases, so does the angle $\theta$. Therefore, since $2 > \frac{1}{2}$, it follows that $\theta_1 > \theta_2$.

Therefore, the straight line with a slope of 2 forms the largest angle with the x-axis.

We will use the formula:

$m=\tan\alpha$

Let's check the slope of minus 6:

$-6=\tan\alpha$

$\tan^{-1}(-6)=\alpha$

$-80.53=\alpha$

$180-80.53=$

$99.47=\alpha_1$

Let's check the slope of one-half:

$\frac{1}{2}=\tan\alpha$

$\tan^{-1}(\frac{1}{2})=\alpha$

$26.56=\alpha_2$

\alpha_1 > \alpha_2

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

• Step 1: Calculate the slope
• Step 2: Apply the point-slope form to get the line equation
• Step 3: Simplify and verify against given choices

Now, let's work through each step:
Step 1: The slope for a line making 135 degrees with the x-axis is $m = \tan(135^\circ) = \tan(135^\circ - 180^\circ) = \tan(-45^\circ) = -1$.
Step 2: Using the point-slope formula $y - y_1 = m(x - x_1)$ with $(x_1, y_1) = (3, 14)$ and $m = -1$, we have:
$y - 14 = -1(x - 3)$.
Simplifying, $y - 14 = -x + 3$.
Rearranging terms gives $y = -x + 17$, or equivalently $y + x = 17$.

Therefore, the equation of the line is $y + x = 17$.

To solve this problem, we'll find the equation of a line passing through $(2,2)$ that makes an angle of $180^\circ$ with the positive x-axis.

• Step 1: Calculate the Slope.
  The slope $m$ of the line can be found using the tangent of the angle. The angle given is $180^\circ$.
  Therefore, $m = \tan(180^\circ) = 0$.
  This tells us that the line is horizontal.
• Step 2: Apply the Point-Slope Form.
  Since the line is horizontal (slope = 0), it has a constant $y$-value.
  The point given is $(2, 2)$, meaning the line's equation is $y = 2$.

Thus, the equation of the line is $y = 2$. This corresponds to choice $4$ in the given options, confirming that it meets all criteria of the problem.

Therefore, the solution to the problem is $y = 2$.

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

• Step 1: Use the formula $m = \tan(\theta)$
• Step 2: Calculate $\tan(100^\circ)$
• Step 3: Determine if the line is ascending or descending based on the slope

Now, let's work through each step:

Step 1: We use the formula for the slope of a line:
$\displaystyle m = \tan(\theta)$

Step 2: Substitute $\theta = 100^\circ$ into the formula:
$\displaystyle m = \tan(100^\circ)$

Using a calculator, $\tan(100^\circ) \approx -5.67$.

Step 3: Since the slope is negative, the line is descending.

Therefore, the solution to the problem is: $m = -5.67$, decreasing.

To solve this problem, let's start by finding the slope $m$ of the line. Since the line makes a 45-degree angle with the positive $x$-axis, the slope $m = \tan(45^\circ) = 1$.

Next, we will use the point-slope form of the line equation, which is given by:

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

where $(x_1, y_1) = (-3, -5)$ and $m = 1$. Substituting these values, we have:

$y + 5 = 1 \cdot (x + 3)$

which simplifies to:

$y + 5 = x + 3$

Subtract 5 from both sides to put it into slope-intercept form $y = mx + b$, we get:

$y = x - 2$

Therefore, the function that represents the line through $(-3, -5)$ with a 45-degree angle with the $x$-axis is $y = x - 2$.