Find a general form of an equation of the line through the point A that satisfies the given condition.
A(8, −1); perpendicular to the line 5x − 5y = 4
the slopes of perpendicular lines are negative-reciprocals
the slope of 5x − 5y = 4 is 1
using point-slope , the A line is ... y + 1 = -1 (x - 8)
To find the equation of a line perpendicular to the given line, we can first find the slope of the given line and then use that information to find the slope of the perpendicular line.
First, let's rearrange the equation of the given line 5x - 5y = 4 into the slope-intercept form y = mx + b by solving for y:
5x - 5y = 4
-5y = -5x + 4
y = (5/5)x - 4/5
y = x - 4/5
From this equation, we can see that the slope of the given line is 1.
The slope of a line perpendicular to the given line is the negative reciprocal of its slope. So, the slope of the perpendicular line will be -1.
Now, let's use the point-slope form of a line to find the equation of the line that passes through point A(8, -1) with slope -1:
y - y1 = m(x - x1)
Substituting the values:
y - (-1) = -1(x - 8)
y + 1 = -x + 8
y = -x + 7
Therefore, the general form of the equation of the line through point A(8, -1) that is perpendicular to the line 5x - 5y = 4 is y = -x + 7.
To find the equation of the line through point A(8, -1) that is perpendicular to the given line, we need to follow these steps:
1. Convert the equation of the given line to slope-intercept form (y = mx + b).
2. Determine the slope of the given line.
3. Determine the slope of the line that is perpendicular to the given line.
4. Use the slope-intercept form (y = mx + b) and the point A(8, -1) to find the equation of the perpendicular line.
Let's proceed step by step:
Step 1: Convert the equation of the given line to slope-intercept form.
Start with the equation: 5x - 5y = 4
Rearrange it to isolate y:
-5y = -5x + 4
Divide through by -5 to get y by itself:
y = (5/5)x - 4/5
Simplify:
y = x - 4/5
So, the given line is in slope-intercept form: y = x - 4/5.
Step 2: Determine the slope of the given line.
From the equation y = x - 4/5, we can see that the coefficient of x, which is 1, represents the slope. So, the slope of the given line is 1.
Step 3: Determine the slope of the line that is perpendicular to the given line.
Two lines are perpendicular when their slopes are negative reciprocals of each other. The negative reciprocal of 1 is -1. Therefore, the slope of the line that is perpendicular to the given line is -1.
Step 4: Use the slope-intercept form (y = mx + b) and the point A(8, -1) to find the equation of the perpendicular line.
Using the point-slope form of a line (y - y₁ = m(x - x₁)) with the slope (-1) and point A(8, -1), we substitute the values:
y - (-1) = -1(x - 8)
y + 1 = -x + 8
Rearrange the equation to slope-intercept form:
y = -x + 7
Therefore, the general form of the equation of the line through point A(8, -1) that is perpendicular to the line 5x - 5y = 4 is y = -x + 7.