Answer :
To find the equation of a line that passes through the point \((8, 4)\) and is parallel to the line \(y = 4x + 2\), we need to follow these steps:
1. Identify the slope of the given line: The given line \(y = 4x + 2\) is in slope-intercept form \(y = mx + b\), where \(m\) is the slope.
- Here, the slope \(m\) is \(4\).
2. Parallel lines have the same slope: Since we need a line parallel to \(y = 4x + 2\), the slope of our new line will also be \(4\).
3. Use the point-slope form of a line equation: The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
- Here, \((x_1, y_1)\) is the point through which the line passes. So, \((x_1, y_1) = (8, 4)\), and \(m = 4\).
4. Substitute the values: Substitute the values \((x_1, y_1)\) and \(m\) into the point-slope form:
[tex]\[ y - 4 = 4(x - 8) \][/tex]
5. Simplify the equation: Distribute the slope and simplify:
[tex]\[ y - 4 = 4x - 32 \][/tex]
Add \(4\) to both sides to isolate \(y\):
[tex]\[ y = 4x - 28 \][/tex]
Thus, the equation of the line that passes through the point \((8, 4)\) and is parallel to the line \(y = 4x + 2\) is \(y = 4x - 28\).
Therefore, the correct option is:
[tex]\[ y = 4x - 28 \][/tex]
1. Identify the slope of the given line: The given line \(y = 4x + 2\) is in slope-intercept form \(y = mx + b\), where \(m\) is the slope.
- Here, the slope \(m\) is \(4\).
2. Parallel lines have the same slope: Since we need a line parallel to \(y = 4x + 2\), the slope of our new line will also be \(4\).
3. Use the point-slope form of a line equation: The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
- Here, \((x_1, y_1)\) is the point through which the line passes. So, \((x_1, y_1) = (8, 4)\), and \(m = 4\).
4. Substitute the values: Substitute the values \((x_1, y_1)\) and \(m\) into the point-slope form:
[tex]\[ y - 4 = 4(x - 8) \][/tex]
5. Simplify the equation: Distribute the slope and simplify:
[tex]\[ y - 4 = 4x - 32 \][/tex]
Add \(4\) to both sides to isolate \(y\):
[tex]\[ y = 4x - 28 \][/tex]
Thus, the equation of the line that passes through the point \((8, 4)\) and is parallel to the line \(y = 4x + 2\) is \(y = 4x - 28\).
Therefore, the correct option is:
[tex]\[ y = 4x - 28 \][/tex]
