Question

1. write an equation in slope-intercept form for
2. identify which lines are parallel: $y = -\frac{1}{2}x + 3$; $y = \frac{1}{2}x + 1$; $y = 2x$; $x + 2y = 4$.
3. identify which lines are perpendicular: $y - 2 = 3x$; $y + 4x = -1$; $y = -\frac{1}{3}x + 5$; $y = \frac{1}{3}x - 4$.
4. write an equation in slope-intercept form for the line that passes through $(0, 6)$ and is parallel to the line described by $y = 2x + 3$.
5. write an equation in slope-intercept form for the line that passes through $(4, 6)$ and is perpendicular to the line described by $y = x - 3$.

Explanation:

Question 15

Step1: Recall slope-intercept form

Slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Step2: Determine the slope of the parallel line

If two lines are parallel, they have the same slope. The line \(y = 2x+3\) has a slope \(m = 2\). So the line we want to find also has a slope \(m = 2\).

Step3: Determine the y - intercept

The line passes through the point \((0,6)\). In the slope - intercept form \(y=mx + b\), when \(x = 0\), \(y=b\). So if the line passes through \((0,6)\), then \(b = 6\).

Step4: Write the equation

Substitute \(m = 2\) and \(b = 6\) into the slope - intercept form \(y=mx + b\). We get \(y=2x + 6\).

Step1: Recall slope - intercept form and perpendicular slope relationship

Slope - intercept form is \(y=mx + b\). If two lines are perpendicular, the product of their slopes is \(- 1\) (i.e., \(m_1\times m_2=-1\)).

Step2: Find the slope of the given line

The line \(y=x - 3\) has a slope \(m_1 = 1\). Let the slope of the perpendicular line be \(m_2\). Then \(1\times m_2=-1\), so \(m_2=-1\).

Step3: Use the point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(4,6)\) and \(m=-1\). Substitute these values into the point - slope form:
\(y - 6=-1(x - 4)\)

Step4: Simplify to slope - intercept form

Expand the right - hand side: \(y-6=-x + 4\). Add 6 to both sides: \(y=-x+4 + 6\), so \(y=-x + 10\).

Answer:

\(y = 2x+6\)

Question 16