You need to use the point slope form of equation of line, such that:
`y - f(x_0) = m_0(x - x_0)`
The problem provides the point ` (x_0,y_0) = (5, - 2)` and the relative parallel position between the line you need to find and the line `2y-x = 5` , hence, you need to use the equation that relates the slopes of the lines, such that:
`m_0 = m`
You may evaluate m converting the given form of the equation of line 2y-x = 5 into the slope intercept form, y = mx + n, such that:
`2y = x + 5 => y = 1/2x + 5/2 => m = 1/2 => m_0 = 1/2`
`y - f(5) = (1/2 )(x - 5) => y + 2 = (1/2)(x - 5) => y = (1/2)*x - 5/2 - 2`
`y = 0.5x - 4.5`
Hence, evaluating the equation of the line, under the given conditions, yields `y = 0.5x - 4.5.`
Two lines are parallel if and only if their slopes are equal or if the ratio of their correspondent coefficients are also equal.
We'll put the given equation into the standard form. For this reason, we'll isolate 2y to the left side:
2y = x + 5
We'll divide by 2:
y = x/2 + 5/2
We'll write the standard form of the equation of the parallel line and we'll identify the value of the slope for both lines.
y = mx + n
m1 = 1/2 and m2 = m
The slopes have like values.
m = 1/2
Now, we'll write the equtaion of the line that has the slope m=1/2 and it passes through the point (5,-2).
y - (-2) = (1/2)(x - 5)
y + 2 = x/2 - 5/2
y = x/2 - 5/2 - 2
y = x/2 - 9/2
2y = x - 9
2y - x + 9 = 0
The equation of the parallel line whose slope is m = 1/2 and it is passing through the point (5,-2) is 2y - x + 9 = 0.