Answer :
To determine the equation of the line parallel to the given equation [tex]\(3x + 2y = 8\)[/tex] that passes through the point [tex]\((-2, 5)\)[/tex]:
### Step-by-Step Solution:
1. Convert the given equation to slope-intercept form [tex]\(y = mx + b\)[/tex]:
The given equation is [tex]\(3x + 2y = 8\)[/tex]. Rearrange it to solve for [tex]\(y\)[/tex]:
[tex]\[ 2y = -3x + 8 \implies y = -\frac{3}{2}x + 4 \][/tex]
2. Identify the slope [tex]\(m\)[/tex]:
From the slope-intercept form [tex]\(y = -\frac{3}{2}x + 4\)[/tex], we can see that the slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
3. Use the point-slope form of the equation to find the equation of the line parallel to the given line and passing through the point [tex]\((-2, 5)\)[/tex]:
The point-slope form 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.
We have:
[tex]\[ x_1 = -2, \quad y_1 = 5, \quad m = -\frac{3}{2} \][/tex]
4. Substitute the known values into the point-slope form:
[tex]\[ y - 5 = -\frac{3}{2}(x + 2) \][/tex]
5. Simplify the equation:
[tex]\[ y - 5 = -\frac{3}{2}x - 3 \][/tex]
[tex]\[ y = -\frac{3}{2}x - 3 + 5 \][/tex]
[tex]\[ y = -\frac{3}{2}x + 2 \][/tex]
### Conclusion:
The equation of the line parallel to [tex]\(3x + 2y = 8\)[/tex] that passes through the point [tex]\((-2, 5)\)[/tex] is:
[tex]\[ y = -\frac{3}{2}x + 2 \][/tex]
Therefore, you should select:
- [tex]\(y = -\frac{3}{2}x + 2\)[/tex]
### Step-by-Step Solution:
1. Convert the given equation to slope-intercept form [tex]\(y = mx + b\)[/tex]:
The given equation is [tex]\(3x + 2y = 8\)[/tex]. Rearrange it to solve for [tex]\(y\)[/tex]:
[tex]\[ 2y = -3x + 8 \implies y = -\frac{3}{2}x + 4 \][/tex]
2. Identify the slope [tex]\(m\)[/tex]:
From the slope-intercept form [tex]\(y = -\frac{3}{2}x + 4\)[/tex], we can see that the slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
3. Use the point-slope form of the equation to find the equation of the line parallel to the given line and passing through the point [tex]\((-2, 5)\)[/tex]:
The point-slope form 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.
We have:
[tex]\[ x_1 = -2, \quad y_1 = 5, \quad m = -\frac{3}{2} \][/tex]
4. Substitute the known values into the point-slope form:
[tex]\[ y - 5 = -\frac{3}{2}(x + 2) \][/tex]
5. Simplify the equation:
[tex]\[ y - 5 = -\frac{3}{2}x - 3 \][/tex]
[tex]\[ y = -\frac{3}{2}x - 3 + 5 \][/tex]
[tex]\[ y = -\frac{3}{2}x + 2 \][/tex]
### Conclusion:
The equation of the line parallel to [tex]\(3x + 2y = 8\)[/tex] that passes through the point [tex]\((-2, 5)\)[/tex] is:
[tex]\[ y = -\frac{3}{2}x + 2 \][/tex]
Therefore, you should select:
- [tex]\(y = -\frac{3}{2}x + 2\)[/tex]
