To find the equation of a line passing through the point [tex]$(-2, 8)$[/tex] and parallel to the line given by [tex]\(2x - 3y - 7 = 0\)[/tex], we will follow these steps:
### 1. Determine the Slope of the Given Line
The equation of the given line is [tex]\(2x - 3y - 7 = 0\)[/tex]. To find its slope, we'll convert it to slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[2x - 3y - 7 = 0 \][/tex]
[tex]\[2x - 7 = 3y\][/tex]
[tex]\[3y = 2x - 7\][/tex]
[tex]\[y = \frac{2}{3}x - \frac{7}{3}\][/tex]
So, the slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{2}{3}\)[/tex].
### 2. Find the Slope of the Parallel Line
Since parallel lines have the same slope, the slope of the new line will also be [tex]\(\frac{2}{3}\)[/tex].
### 3. Use the Point-Slope Form to Write the Equation
The point-slope form of a line's equation is given by:
[tex]\[y - y_1 = m(x - x_1)\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope. Here, [tex]\((x_1, y_1) = (-2, 8)\)[/tex] and [tex]\(m = \frac{2}{3}\)[/tex]. Substituting these values in,
[tex]\[y - 8 = \frac{2}{3}(x + 2)\][/tex]
### 4. Convert to General Form
Now we'll simplify and convert this to the general form [tex]\(Ax + By + C = 0\)[/tex]:
[tex]\[y - 8 = \frac{2}{3}(x + 2)\][/tex]
[tex]\[y - 8 = \frac{2}{3}x + \frac{4}{3}\][/tex]
To clear the fraction, multiply every term by 3:
[tex]\[3(y - 8) = 2(x + 2)\][/tex]
[tex]\[3y - 24 = 2x + 4\][/tex]
Now, rearrange terms to get the equation in general form:
[tex]\[2x - 3y + 28 = 0\][/tex]
So the equations of the line are:
- Point-slope form: [tex]\( y - 8 = \frac{2}{3}(x + 2) \)[/tex].
- General form: [tex]\( 2x - 3y + 28 = 0 \)[/tex].
