Answer :
To determine which of the given equations are correct for the line passing through the point [tex]\((-3, 2)\)[/tex] with a slope [tex]\(m = \frac{2}{3}\)[/tex], we need to check each equation step by step.
### Step-by-Step Solution:
1. Point-Slope Form:
The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Given the point [tex]\((x_1, y_1) = (-3, 2)\)[/tex] and slope [tex]\(m = \frac{2}{3}\)[/tex], plugging these values into the point-slope form, we get:
[tex]\[ y - 2 = \frac{2}{3}(x - (-3)) \][/tex]
Simplifying the equation:
[tex]\[ y - 2 = \frac{2}{3}(x + 3) \][/tex]
Therefore, the equation [tex]\(y - 2 = \frac{2}{3}(x + 3)\)[/tex] is correct. This corresponds to option D.
2. Slope-Intercept Form:
The slope-intercept form of the equation of a line is:
[tex]\[ y = mx + b \][/tex]
We need to convert the point-slope form to the slope-intercept form by isolating [tex]\(y\)[/tex]:
[tex]\[ y - 2 = \frac{2}{3}(x + 3) \][/tex]
Distributing [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ y - 2 = \frac{2}{3}x + \frac{2}{3} \cdot 3 \][/tex]
[tex]\[ y - 2 = \frac{2}{3}x + 2 \][/tex]
Adding 2 to both sides:
[tex]\[ y = \frac{2}{3}x + 4 \][/tex]
Therefore, the equation [tex]\(y = \frac{2}{3}x + 4\)[/tex] is correct. This corresponds to option B.
3. Standard Form (Verifying Option A):
Option A is given in the standard form. We need to convert it to the slope-intercept form to verify its correctness:
[tex]\[ 2x - 3y = -12 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ -3y = -2x - 12 \][/tex]
Dividing by -3:
[tex]\[ y = \frac{2}{3}x + 4 \][/tex]
This matches the slope-intercept form we derived earlier. Therefore, the equation [tex]\(2x - 3y = -12\)[/tex] is correct. This corresponds to option A.
4. Verifying Option C:
The equation in option C is given by:
[tex]\[ y = \frac{2}{3}x \][/tex]
Substitute [tex]\(x = -3\)[/tex] into the equation to check if it satisfies the given point [tex]\((-3, 2)\)[/tex]:
[tex]\[ y = \frac{2}{3}(-3) \][/tex]
[tex]\[ y = -2 \][/tex]
Since [tex]\(-2\)[/tex] is not equal to [tex]\(2\)[/tex], the equation [tex]\(y = \frac{2}{3}x\)[/tex] does not pass through the point [tex]\((-3, 2)\)[/tex]. Therefore, option C is incorrect.
### Conclusion:
The correct equations for the line passing through the point [tex]\((-3, 2)\)[/tex] with a slope of [tex]\(\frac{2}{3}\)[/tex] are:
- Option A: [tex]\(2x - 3y = -12\)[/tex]
- Option B: [tex]\(y = \frac{2}{3}x + 4\)[/tex]
- Option D: [tex]\(y - 2 = \frac{2}{3}(x + 3)\)[/tex]
Thus, the correct options are:
[tex]\[ \text{Options: } [\text{A, B, D}] \][/tex]
### Step-by-Step Solution:
1. Point-Slope Form:
The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Given the point [tex]\((x_1, y_1) = (-3, 2)\)[/tex] and slope [tex]\(m = \frac{2}{3}\)[/tex], plugging these values into the point-slope form, we get:
[tex]\[ y - 2 = \frac{2}{3}(x - (-3)) \][/tex]
Simplifying the equation:
[tex]\[ y - 2 = \frac{2}{3}(x + 3) \][/tex]
Therefore, the equation [tex]\(y - 2 = \frac{2}{3}(x + 3)\)[/tex] is correct. This corresponds to option D.
2. Slope-Intercept Form:
The slope-intercept form of the equation of a line is:
[tex]\[ y = mx + b \][/tex]
We need to convert the point-slope form to the slope-intercept form by isolating [tex]\(y\)[/tex]:
[tex]\[ y - 2 = \frac{2}{3}(x + 3) \][/tex]
Distributing [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ y - 2 = \frac{2}{3}x + \frac{2}{3} \cdot 3 \][/tex]
[tex]\[ y - 2 = \frac{2}{3}x + 2 \][/tex]
Adding 2 to both sides:
[tex]\[ y = \frac{2}{3}x + 4 \][/tex]
Therefore, the equation [tex]\(y = \frac{2}{3}x + 4\)[/tex] is correct. This corresponds to option B.
3. Standard Form (Verifying Option A):
Option A is given in the standard form. We need to convert it to the slope-intercept form to verify its correctness:
[tex]\[ 2x - 3y = -12 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ -3y = -2x - 12 \][/tex]
Dividing by -3:
[tex]\[ y = \frac{2}{3}x + 4 \][/tex]
This matches the slope-intercept form we derived earlier. Therefore, the equation [tex]\(2x - 3y = -12\)[/tex] is correct. This corresponds to option A.
4. Verifying Option C:
The equation in option C is given by:
[tex]\[ y = \frac{2}{3}x \][/tex]
Substitute [tex]\(x = -3\)[/tex] into the equation to check if it satisfies the given point [tex]\((-3, 2)\)[/tex]:
[tex]\[ y = \frac{2}{3}(-3) \][/tex]
[tex]\[ y = -2 \][/tex]
Since [tex]\(-2\)[/tex] is not equal to [tex]\(2\)[/tex], the equation [tex]\(y = \frac{2}{3}x\)[/tex] does not pass through the point [tex]\((-3, 2)\)[/tex]. Therefore, option C is incorrect.
### Conclusion:
The correct equations for the line passing through the point [tex]\((-3, 2)\)[/tex] with a slope of [tex]\(\frac{2}{3}\)[/tex] are:
- Option A: [tex]\(2x - 3y = -12\)[/tex]
- Option B: [tex]\(y = \frac{2}{3}x + 4\)[/tex]
- Option D: [tex]\(y - 2 = \frac{2}{3}(x + 3)\)[/tex]
Thus, the correct options are:
[tex]\[ \text{Options: } [\text{A, B, D}] \][/tex]
