Answer :
To solve the problem of finding the equation of a line that is parallel to the line \(5x + 2y = 12\) and passes through the point \((-2, 4)\), follow these steps:
### Step-by-Step Solution
1. Convert the Given Line Equation to Slope-Intercept Form:
Start by converting the given line equation from standard form \(5x + 2y = 12\) to slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
[tex]\[ 5x + 2y = 12 \][/tex]
Subtract \(5x\) from both sides:
[tex]\[ 2y = -5x + 12 \][/tex]
Divide every term by 2:
[tex]\[ y = -\frac{5}{2}x + 6 \][/tex]
So, the slope (\(m\)) of the given line is:
[tex]\[ m = -\frac{5}{2} \][/tex]
2. Find the Slope of the Parallel Line:
Parallel lines share the same slope. Thus, the slope of the line that is parallel to the given line and passes through the point \((-2, 4)\) is also \(m = -\frac{5}{2}\).
3. Form the Equation of the Parallel Line:
Use the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.
Here, \((x_1, y_1) = (-2, 4)\) and \(m = -\frac{5}{2}\).
Substitute these values into the point-slope form:
[tex]\[ y - 4 = -\frac{5}{2}(x + 2) \][/tex]
4. Simplify the Equation:
Distribute the slope \(-\frac{5}{2}\) through the parentheses:
[tex]\[ y - 4 = -\frac{5}{2}x - \frac{5}{2} \cdot 2 \][/tex]
Simplify:
[tex]\[ y - 4 = -\frac{5}{2}x - 5 \][/tex]
Add 4 to both sides to solve for \(y\):
[tex]\[ y = -\frac{5}{2}x - 5 + 4 \][/tex]
[tex]\[ y = -\frac{5}{2}x - 1 \][/tex]
Hence, the equation of the line that is parallel to \(5x + 2y = 12\) and passes through the point \((-2, 4)\) is:
[tex]\[ \boxed{y = -\frac{5}{2}x - 1} \][/tex]
So the correct answer is:
[tex]\( y = -\frac{5}{2}x - 1 \)[/tex]
### Step-by-Step Solution
1. Convert the Given Line Equation to Slope-Intercept Form:
Start by converting the given line equation from standard form \(5x + 2y = 12\) to slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
[tex]\[ 5x + 2y = 12 \][/tex]
Subtract \(5x\) from both sides:
[tex]\[ 2y = -5x + 12 \][/tex]
Divide every term by 2:
[tex]\[ y = -\frac{5}{2}x + 6 \][/tex]
So, the slope (\(m\)) of the given line is:
[tex]\[ m = -\frac{5}{2} \][/tex]
2. Find the Slope of the Parallel Line:
Parallel lines share the same slope. Thus, the slope of the line that is parallel to the given line and passes through the point \((-2, 4)\) is also \(m = -\frac{5}{2}\).
3. Form the Equation of the Parallel Line:
Use the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.
Here, \((x_1, y_1) = (-2, 4)\) and \(m = -\frac{5}{2}\).
Substitute these values into the point-slope form:
[tex]\[ y - 4 = -\frac{5}{2}(x + 2) \][/tex]
4. Simplify the Equation:
Distribute the slope \(-\frac{5}{2}\) through the parentheses:
[tex]\[ y - 4 = -\frac{5}{2}x - \frac{5}{2} \cdot 2 \][/tex]
Simplify:
[tex]\[ y - 4 = -\frac{5}{2}x - 5 \][/tex]
Add 4 to both sides to solve for \(y\):
[tex]\[ y = -\frac{5}{2}x - 5 + 4 \][/tex]
[tex]\[ y = -\frac{5}{2}x - 1 \][/tex]
Hence, the equation of the line that is parallel to \(5x + 2y = 12\) and passes through the point \((-2, 4)\) is:
[tex]\[ \boxed{y = -\frac{5}{2}x - 1} \][/tex]
So the correct answer is:
[tex]\( y = -\frac{5}{2}x - 1 \)[/tex]
