$$ y=-4 x-5 $$ yhe line with slope 5 that passes through $$ (2,-5) $$

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UpStudy AI · Accepted Answer

To find the equation of a line with a slope of $$ 5 $$ that passes through the point $$ (2, -5) $$, you can use the **point-slope form** of a linear equation. Here's how you can do it step by step:

### Step 1: Recall the 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.

### Step 2: Plug in the Given Values
From your question:
- Slope ($$ m $$) = $$ 5 $$
- Point ($$ x_1, y_1 $$) = $$ (2, -5) $$

Substitute these values into the point-slope formula:
$$
y - (-5) = 5(x - 2)
$$
$$
y + 5 = 5(x - 2)
$$

### Step 3: Simplify the Equation
Distribute the slope on the right side:
$$
y + 5 = 5x - 10
$$

Subtract $$ 5 $$ from both sides to solve for $$ y $$:
$$
y = 5x - 15
$$

### Final Equation
The equation of the line with a slope of $$ 5 $$ that passes through the point $$ (2, -5) $$ is:
$$
\boxed{y = 5x - 15}
$$

### Additional Context (Optional)
If you also have the line $$ y = -4x - 5 $$ in your problem, and you’re looking to analyze the relationship between these two lines (such as finding their intersection point), you can set them equal to each other:

$$
5x - 15 = -4x - 5
$$

Solving for $$ x $$:
$$
5x + 4x = -5 + 15
$$
$$
9x = 10 \implies x = \frac{10}{9}
$$

Plugging $$ x = \frac{10}{9} $$ back into one of the equations to find $$ y $$:
$$
y = 5\left(\frac{10}{9}\right) - 15 = \frac{50}{9} - \frac{135}{9} = -\frac{85}{9}
$$

So, the two lines intersect at:
$$
\left( \frac{10}{9}, -\frac{85}{9} \right)
$$

But if your primary goal was to find the equation of the line with slope $$ 5 $$ through $$ (2, -5) $$, the answer is $$ y = 5x - 15 $$.
The equation of the line with a slope of 5 that passes through the point (2, -5) is $$ y = 5x - 15 $$.

\( y=-4 x-5 \) yhe line with slope 5 that passes through \( (2,-5) \)

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