Question

5 determine whether the lines $x + 2y - 4 = 0$ and $2x - y - 4 = 0$ are parallel lines.

my calculations

6 find the equation of the line that passes through the point $a(3,5)$ and is parallel to the line $y = x - 4$.

my calculations

Explanation:

Question 5

Step1: Convert to slope - intercept form

For the line \(x + 2y-4 = 0\), solve for \(y\):
\(2y=-x + 4\), then \(y=-\frac{1}{2}x + 2\). The slope \(m_1=-\frac{1}{2}\).

For the line \(2x - y-4 = 0\), solve for \(y\):
\(y = 2x-4\). The slope \(m_2 = 2\).

Step2: Compare slopes

Parallel lines have equal slopes. Since \(m_1=-\frac{1}{2}\) and \(m_2 = 2\), and \(m_1
eq m_2\).

Step1: Determine the slope of the parallel line

The line \(y=x - 4\) is in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept). So its slope \(m = 1\).
If two lines are parallel, they have the same slope. So the line passing through \(A(3,5)\) also has a slope \(m = 1\).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(3,5)\) and \(m = 1\).
Substitute the values: \(y - 5=1\times(x - 3)\).

Step3: Simplify the equation

\(y-5=x - 3\), then \(y=x - 3 + 5\), so \(y=x + 2\).

Answer:

The lines \(x + 2y - 4=0\) and \(2x - y - 4 = 0\) are not parallel.

Question 6