Answer :
Let's solve the problem step-by-step.
### Step 1: Find the slope of the original line
Given that the line passes through the point [tex]\((-6, -2)\)[/tex] and has a [tex]\(y\)[/tex]-intercept of [tex]\((0, 1)\)[/tex], we need to find the slope of the original line. The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((-6, -2)\)[/tex] and [tex]\((0, 1)\)[/tex]:
[tex]\[ m = \frac{1 - (-2)}{0 - (-6)} = \frac{1 + 2}{0 + 6} = \frac{3}{6} = \frac{1}{2} \][/tex]
So, the slope of the original line is [tex]\(\frac{1}{2}\)[/tex].
### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original line’s slope. If the slope of the original line is [tex]\(m\)[/tex], then the slope of the perpendicular line [tex]\(m_\perp\)[/tex] is:
[tex]\[ m_\perp = -\frac{1}{m} \][/tex]
Substituting the slope of the original line, [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ m_\perp = -\frac{1}{\frac{1}{2}} = -2 \][/tex]
### Step 3: Use the point-slope form to find the equation of the perpendicular line
The perpendicular line passes through the point [tex]\((2, 3)\)[/tex] and has a slope of [tex]\(-2\)[/tex]. The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting [tex]\(m = -2\)[/tex] and the point [tex]\((2, 3)\)[/tex]:
[tex]\[ y - 3 = -2(x - 2) \][/tex]
### Step 4: Simplify to the slope-intercept form [tex]\(y = mx + b\)[/tex]
Distribute the slope and simplify:
[tex]\[ y - 3 = -2x + 4 \][/tex]
Add 3 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -2x + 7 \][/tex]
### Conclusion
The equation of the line that is perpendicular to the original line and passes through the point [tex]\((2, 3)\)[/tex] is:
[tex]\[ y = -2x + 7 \][/tex]
Thus, the correct answer is:
D. [tex]\(y = -2x + 7\)[/tex]
### Step 1: Find the slope of the original line
Given that the line passes through the point [tex]\((-6, -2)\)[/tex] and has a [tex]\(y\)[/tex]-intercept of [tex]\((0, 1)\)[/tex], we need to find the slope of the original line. The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((-6, -2)\)[/tex] and [tex]\((0, 1)\)[/tex]:
[tex]\[ m = \frac{1 - (-2)}{0 - (-6)} = \frac{1 + 2}{0 + 6} = \frac{3}{6} = \frac{1}{2} \][/tex]
So, the slope of the original line is [tex]\(\frac{1}{2}\)[/tex].
### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original line’s slope. If the slope of the original line is [tex]\(m\)[/tex], then the slope of the perpendicular line [tex]\(m_\perp\)[/tex] is:
[tex]\[ m_\perp = -\frac{1}{m} \][/tex]
Substituting the slope of the original line, [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ m_\perp = -\frac{1}{\frac{1}{2}} = -2 \][/tex]
### Step 3: Use the point-slope form to find the equation of the perpendicular line
The perpendicular line passes through the point [tex]\((2, 3)\)[/tex] and has a slope of [tex]\(-2\)[/tex]. The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting [tex]\(m = -2\)[/tex] and the point [tex]\((2, 3)\)[/tex]:
[tex]\[ y - 3 = -2(x - 2) \][/tex]
### Step 4: Simplify to the slope-intercept form [tex]\(y = mx + b\)[/tex]
Distribute the slope and simplify:
[tex]\[ y - 3 = -2x + 4 \][/tex]
Add 3 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -2x + 7 \][/tex]
### Conclusion
The equation of the line that is perpendicular to the original line and passes through the point [tex]\((2, 3)\)[/tex] is:
[tex]\[ y = -2x + 7 \][/tex]
Thus, the correct answer is:
D. [tex]\(y = -2x + 7\)[/tex]
