To determine the standard form of the equation of the line that passes through the point [tex]\(A(5,9)\)[/tex] and is parallel to the line [tex]\(y = 5x - 9\)[/tex], follow these steps:
1. Identify the Slope:
- The given line [tex]\(y = 5x - 9\)[/tex] is in slope-intercept form [tex]\(y = mx + b\)[/tex], where the slope [tex]\(m = 5\)[/tex].
- Since parallel lines have the same slope, the slope of our line will also be [tex]\(5\)[/tex].
2. Use the Point-Slope Form:
- The point-slope form of the equation of a line is given by [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.
- Plug in the slope [tex]\(m = 5\)[/tex] and the point [tex]\(A(5,9)\)[/tex]:
[tex]\[
y - 9 = 5(x - 5)
\][/tex]
3. Simplify to Slope-Intercept Form:
- Distribute the slope [tex]\(5\)[/tex] on the right side:
[tex]\[
y - 9 = 5x - 25
\][/tex]
- Add [tex]\(9\)[/tex] to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = 5x - 16
\][/tex]
4. Convert to Standard Form:
- The standard form of a line is [tex]\(Ax + By = C\)[/tex], where [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] are integers, and [tex]\(A\)[/tex] should be positive.
- Starting with the equation [tex]\(y = 5x - 16\)[/tex], subtract [tex]\(5x\)[/tex] from both sides to move the [tex]\(x\)[/tex]-term to the left:
[tex]\[
-5x + y = -16
\][/tex]
- Multiply the entire equation by [tex]\(-1\)[/tex] to make the coefficient of [tex]\(x\)[/tex] positive:
[tex]\[
5x - y = 16
\][/tex]
Thus, the standard form of the equation of the line that passes through [tex]\(A(5, 9)\)[/tex] and is parallel to the line [tex]\(y = 5x - 9\)[/tex] is:
[tex]\[
5x - y = 16
\][/tex]
