To find the equation of the line through point [tex]\((-4, -4)\)[/tex] and perpendicular to the line given by [tex]\(2x + 3y = 3\)[/tex], we'll follow these steps:
1. Determine the slope of the given line:
- The given line's equation is [tex]\(2x + 3y = 3\)[/tex]. To find its slope, rearrange it into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
- Start by isolating [tex]\(y\)[/tex]:
[tex]\[ 3y = -2x + 3 \][/tex]
[tex]\[ y = \frac{-2}{3} x + 1 \][/tex]
- Hence, the slope ([tex]\(m_1\)[/tex]) of the given line is [tex]\(-2/3\)[/tex].
2. Find the slope of the line perpendicular to the given line:
- The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope ([tex]\(m_2\)[/tex]) of the line perpendicular to the given line is:
[tex]\[ m_2 = -\frac{1}{-2/3} = \frac{3}{2} \][/tex]
3. Use the point-slope form to write the equation of the perpendicular line:
- The point-slope form of a line 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, the point is [tex]\((-4, -4)\)[/tex] and the slope ([tex]\(m_2\)[/tex]) is [tex]\(3/2\)[/tex].
- Plug these values into the point-slope form:
[tex]\[ y - (-4) = \frac{3}{2} (x - (-4)) \][/tex]
Simplify:
[tex]\[ y + 4 = \frac{3}{2}(x + 4) \][/tex]
4. Simplify to the slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Distribute the slope [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[ y + 4 = \frac{3}{2}x + \frac{3}{2} \cdot 4 \][/tex]
[tex]\[ y + 4 = \frac{3}{2}x + 6 \][/tex]
- Subtract 4 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{3}{2} x + 6 - 4 \][/tex]
[tex]\[ y = \frac{3}{2} x + 2 \][/tex]
So, the equation of the line through point [tex]\((-4, -4)\)[/tex] and perpendicular to the line [tex]\(2x + 3y = 3\)[/tex] is:
[tex]\[
y = \frac{3}{2} x + 2
\][/tex]
