Certainly! Let's break down the problem step by step.
To find the equation of a line that is perpendicular to the given line, we need to determine its slope. We can do this by using the fact that perpendicular lines have slopes that are negative reciprocals of each other.
The given line is in point-slope form: y - 3 = 3(x + 1). We can rewrite it in slope-intercept form (y = mx + b) by simplifying it:
y - 3 = 3x + 3
y = 3x + 6
From this form, we can see that the slope of the given line is 3.
To find the slope of the perpendicular line, we take the negative reciprocal of 3. The negative reciprocal of a number is obtained by flipping its fraction and changing its sign:
Slope of the given line = 3
Slope of the perpendicular line = -(1/3)
Therefore, the slope of the perpendicular line is -1/3.
Next, we can use the slope-intercept form (y = mx + b) of the perpendicular line and the given point (5, 1) to find the equation in standard form.
Using the slope-intercept form, we have: y - y1 = m(x - x1)
y - 1 = -(1/3)(x - 5)
y - 1 = -(1/3)x + 5/3
Now, let's convert the equation to standard form by rearranging the variables and constants:
3(y - 1) = -x + 5
3y - 3 = -x + 5
x + 3y = 8
Hence, the equation of the perpendicular line in standard form is x + 3y = 8.
To find the x-intercept of this line, we set y to zero (since the x-intercept occurs when y is zero) and solve for x:
x + 3(0) = 8
x = 8
Therefore, the x-intercept of the perpendicular line is 8.
I hope this explanation helps you understand the process! If you have any further questions, feel free to ask.