Question

practice final exam
120 points possible 10/20 answered
question 11
find the equation of the line that is parallel to the line 2x + 4y = 5 and which passes through the point (5, - 4.5).

Explanation:

Step1: Find slope of given line

Rewrite \(2x + 4y = 5\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
Subtract \(2x\) from both sides: \(4y=-2x + 5\).
Divide by 4: \(y=-\frac{2}{4}x+\frac{5}{4}=-\frac{1}{2}x+\frac{5}{4}\).
The slope of the given line is \(m =-\frac{1}{2}\). Parallel lines have the same slope, so the slope of the required line is also \(m =-\frac{1}{2}\).

Step2: Use point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(5,-4.5)\) and \(m =-\frac{1}{2}\).
Substitute the values: \(y-(-4.5)=-\frac{1}{2}(x - 5)\).
Simplify: \(y + 4.5=-\frac{1}{2}x+\frac{5}{2}\).
Since \(4.5=\frac{9}{2}\), we have \(y+\frac{9}{2}=-\frac{1}{2}x+\frac{5}{2}\).
Subtract \(\frac{9}{2}\) from both sides: \(y=-\frac{1}{2}x+\frac{5}{2}-\frac{9}{2}\).
\(y=-\frac{1}{2}x-\frac{4}{2}=-\frac{1}{2}x - 2\).
We can also write it in standard form. Multiply both sides by 2: \(2y=-x - 4\), then \(x+2y=-4\) or \(2y+x=-4\) (or in slope - intercept form \(y =-\frac{1}{2}x-2\)).

Answer:

The equation of the line is \(y=-\frac{1}{2}x - 2\) (or \(x + 2y=-4\))