To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then take the negative reciprocal of that slope.
The given line is in the form 4x + 5y = 4. We need to rewrite it in the form y = mx + b.
To do this, let's isolate y by subtracting 4x from both sides:
5y = -4x + 4
Next, divide each term by 5 to solve for y:
y = (-4/5)x + 4/5
So, the slope of the given line is -4/5.
Since the line we are looking for is perpendicular to the given line, its slope will be the negative reciprocal of -4/5.
To find the negative reciprocal, flip the fraction and change its sign:
m = -(5/4)
Now, we have the slope for the line that passes through the point (7,8).
To find the y-intercept (b), we can use the point-slope form of a line:
y - y₁ = m(x - x₁)
Substituting the values of (7,8) and m = -(5/4):
y - 8 = -(5/4)(x - 7)
Next, distribute -(5/4) to the terms inside the parentheses:
y - 8 = -(5/4)x + 35/4
Now, let's isolate y by adding 8 to both sides:
y = -(5/4)x + 35/4 + 8
Combining the terms on the right-hand side:
y = -(5/4)x + 35/4 + 32/4
Simplifying:
y = -(5/4)x + 67/4
Therefore, the equation of the line that passes through the point (7,8) and is perpendicular to the line 4x + 5y = 4 can be written as y = -(5/4)x + 67/4. Thus, the values of m and b are -(5/4) and 67/4, respectively.