To find the standard form of the equation of a line passing through the point [tex]\( (5, -29) \)[/tex] with a y-intercept of [tex]\( 1 \)[/tex], follow these steps:
1. Identify the given information:
- The point [tex]\( (x_1, y_1) = (5, -29) \)[/tex]
- The y-intercept [tex]\( b = 1 \)[/tex]
2. Find the slope (m) of the line:
The y-intercept means the line crosses the y-axis at [tex]\( (0, 1) \)[/tex].
To find the slope, use the formula for the slope between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\( (x_2, y_2) = (0,1) \)[/tex] and [tex]\( (x_1, y_1) = (5, -29) \)[/tex]:
[tex]\[
m = \frac{1 - (-29)}{0 - 5} = \frac{30}{-5} = -6
\][/tex]
3. Write the equation in slope-intercept form:
Using the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = -6x + 1
\][/tex]
4. Convert the equation to standard form [tex]\( Ax + By = C \)[/tex]:
Rearrange the equation to get all terms involving [tex]\( x \)[/tex] and [tex]\( y \)[/tex] on the left side and the constant on the right side:
[tex]\[
y = -6x + 1 \implies 6x + y = 1
\][/tex]
5. Standard form:
The standard form of the equation is
[tex]\[
6x + y = 1
\][/tex]
Thus, the standard form of the equation of the line passing through the point [tex]\( (5, -29) \)[/tex] with a y-intercept of 1 is [tex]\( 6x + y = 1 \)[/tex].
