slope of $$ \frac{1}{5} $$ and through $$ (-5,8) $$

Question

UpStudy AI · Accepted Answer

To find the equation of a line with a slope of $$ \frac{1}{5} $$ that passes through the point $$ (-5, 8) $$, you can use the **point-slope form** of a linear equation. Here's a step-by-step guide:

### 1. Point-Slope Form
The point-slope form of a line is given by:

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

Where:
- $$ m $$ is the slope of the line.
- $$ (x_1, y_1) $$ is a point on the line.

### 2. Plug in the Given Values
Given:
- Slope $$ m = \frac{1}{5} $$
- Point $$ (x_1, y_1) = (-5, 8) $$

Substitute these values into the point-slope form:

$$
y - 8 = \frac{1}{5}(x - (-5))
$$
$$
y - 8 = \frac{1}{5}(x + 5)
$$

### 3. Simplify to Slope-Intercept Form
To express the equation in the slope-intercept form $$ y = mx + b $$, where $$ b $$ is the y-intercept, follow these steps:

$$
y - 8 = \frac{1}{5}x + \frac{1}{5} \times 5
$$
$$
y - 8 = \frac{1}{5}x + 1
$$
$$
y = \frac{1}{5}x + 1 + 8
$$
$$
y = \frac{1}{5}x + 9
$$

### 4. Final Equation
The equation of the line in slope-intercept form is:

$$
\boxed{y = \frac{1}{5}x + 9}
$$

### 5. Alternative Forms

- **Standard Form:** $$ Ax + By = C $$

To convert to standard form:

$$
  y = \frac{1}{5}x + 9
  $$
  
  Multiply all terms by 5 to eliminate the fraction:

$$
  5y = x + 45
  $$
  
  Rearrange:

$$
  x - 5y + 45 = 0 \quad \text{or} \quad x - 5y = -45
  $$
  
  Typically, standard form is written with positive $$ A $$:

$$
  \boxed{x - 5y = -45}
  $$

- **Point-Slope Form:**

As derived earlier:

$$
  \boxed{y - 8 = \frac{1}{5}(x + 5)}
  $$

### 6. Graphical Representation

If you plot the line $$ y = \frac{1}{5}x + 9 $$:

- **Slope $$ \frac{1}{5} $$:** For every 5 units you move to the right (positive direction on the x-axis), the line rises by 1 unit.
- **Y-Intercept $$ 9 $$:** The line crosses the y-axis at the point $$ (0, 9) $$.

### 7. Verification

Ensure that the point $$ (-5, 8) $$ lies on the line:

$$
y = \frac{1}{5}x + 9
$$
$$
8 = \frac{1}{5}(-5) + 9
$$
$$
8 = -1 + 9
$$
$$
8 = 8 \quad \checkmark
$$

The calculation confirms that the point $$ (-5, 8) $$ satisfies the equation.

---

Feel free to ask if you need further assistance or explanations!
The equation of the line with a slope of $$ \frac{1}{5} $$ passing through the point $$ (-5, 8) $$ is: $$ y = \frac{1}{5}x + 9 $$

slope of \( \frac{1}{5} \) and through \( (-5,8) \)

Real Tutor Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions