To find the equation of a line parallel to [tex]\( y = \frac{5}{4}x + c \)[/tex] that passes through the point [tex]\((-1, 1)\)[/tex], we need to follow these steps:
1. Determine the Slope:
The slope of the given line [tex]\( y = \frac{5}{4}x + c \)[/tex] is [tex]\(\frac{5}{4}\)[/tex]. Since parallel lines have the same slope, the slope of our desired line is also [tex]\(\frac{5}{4}\)[/tex].
2. Use the Point-Slope Form:
The point-slope form of a line is given by the equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line. Plugging in the slope [tex]\(\frac{5}{4}\)[/tex] and the point [tex]\((-1, 1)\)[/tex]:
[tex]\[
y - 1 = \frac{5}{4}(x + 1)
\][/tex]
3. Simplify the Point-Slope Form to the Standard Form:
We transform the equation to the standard form [tex]\( Ax + By = C \)[/tex].
First, distribute the slope on the right-hand side:
[tex]\[
y - 1 = \frac{5}{4}x + \frac{5}{4}
\][/tex]
Multiply every term by 4 to clear the fraction:
[tex]\[
4(y - 1) = 5x + 5
\][/tex]
Simplify and rearrange:
[tex]\[
4y - 4 = 5x + 5
\][/tex]
[tex]\[
4y - 5x = 9
\][/tex]
Finally, rearrange to get the form [tex]\( Ax + By = C \)[/tex]:
[tex]\[
5x - 4y = -9
\][/tex]
So, the coefficients are:
[tex]\( A = 5 \)[/tex],
[tex]\( B = -4 \)[/tex],
[tex]\( C = -9 \)[/tex].
Thus, the equation of the line is:
[tex]\[ 5x - 4y = -9 \][/tex]
