We want to find the point of intersection of the given lines. Since we do not know their equations, we have to start by finding them.
Equation of the Perpendicular Line
Two lines are perpendicular when their slopes are negative reciprocals. This means that the product of a given slope and the slope of a line perpendicular to it will be -1.
m_1* m_2=-1
Since the given equation is not written in slope-intercept form, we will have to rewrite it before identifying the slope.
Looking at the equation, we can see that the slope of the line is 6.
y= 6x-7If we substitute this value into our negative reciprocal equation for m_1, we can solve for the slope of a perpendicular line, m_2.
Any line perpendicular to the given equation will have a slope of - 16.
Writing the Equation of the Perpendicular Line
Using the slope m_2= - 16, we can write the general equation of all lines perpendicular to the given equation in slope-intercept form.
y= -1/6x+b
Let's substitute the given point ( 0, 6) into this equation for x and y and solve for the y-intercept b of the perpendicular line.
Now that we have the y-intercept, we can complete the equation. The line given by this equation both is perpendicular to 6x-y=7 and passes through the point (0,6).
y= -1/6x+6
Equation of the Parallel Line
Now we need to find the equation of a line parallel to the following one.
y= 2/3x-4
When lines are parallel, they have the same slope. Because of this, we know that all lines that are parallel to this line will have a slope of 23.
We can write a general equation in slope-intercept form for these lines.
y= 2/3x+ b
We are asked to write the equation of a line parallel to the one with given equation that passes through the point ( - 3, -1). Let's substitute this point into the equation for x and y and solve for the y-intercept b of the parallel line.
Now that we have the y-intercept, we can write the line parallel to y= 23x-4 that passes through (- 3,-1).
y= 2/3x+ 1
Finding the Point of Intersection
If we graph the given equations, we can determine the number of solutions to the system. This will be the point at which the lines intersect. We found both the slope m and y-intercept b when we were finding the equations of our lines.
Equation
Slope m
y-intercept b
y= -1/6x+ 6
-1/6
(0, 6)
y= 2/3x+ 1
2/3
(0, 1)
We will start graphing the system by plotting the y-intercepts of the equations. We can then use the slopes to determine another point that satisfies each equation and connect the points with a line.
We can see that the lines intersect at exactly one point.
The point of intersection at (6, 5) is the only solution to the system.
