To find the equation of the line that is parallel to 2y = 3(2 - 3x) and passes through the point of intersection of the lines y = x + 8 and y = -3x + 4, we need to follow these steps:
Step 1: Find the point of intersection of the lines y = x + 8 and y = -3x + 4.
Step 2: Determine the slope of the line 2y = 3(2 - 3x) (which is parallel to the desired line).
Step 3: Use the slope from step 2 and the point of intersection from step 1 to find the equation of the desired line using the point-slope form.
Let's go through each step:
Step 1: Find the point of intersection of the lines y = x + 8 and y = -3x + 4.
To find the point of intersection, we need to solve the two equations simultaneously:
y = x + 8 ...(Equation 1)
y = -3x + 4 ...(Equation 2)
We can set Equation 1 equal to Equation 2:
x + 8 = -3x + 4
Now, solve for x:
4x = -4
x = -1
Substitute the value of x into either Equation 1 or Equation 2 to find the corresponding y-value:
y = -1 + 8
y = 7
So, the point of intersection is (-1, 7).
Step 2: Determine the slope of the line 2y = 3(2 - 3x).
The given equation is 2y = 3(2 - 3x). We can rewrite it in slope-intercept form (y = mx + b) by dividing both sides by 2:
y = (3(2 - 3x))/2
y = (6 - 9x)/2
y = 3 - (9/2)x
The slope of this line is -(9/2).
Step 3: Use the slope from step 2 and the point of intersection from step 1 to find the equation of the desired line using the point-slope form.
The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
Using the point (-1, 7) and slope -(9/2), we can write the equation as:
y - 7 = -(9/2)(x + 1)
Now, simplify and rewrite it in the slope-intercept form:
y - 7 = -(9/2)x - 9/2
y = -(9/2)x - 9/2 + 7
y = -(9/2)x + 5/2
Therefore, the equation of the line parallel to 2y = 3(2 - 3x) and passing through the point of intersection of the lines y = x + 8 and y = -3x + 4 is y = -(9/2)x + 5/2.