Let's break down and solve each part of the given problem.

Problem:

Find the equation for the line that:

a) Is parallel to the line defined by $y = 2x + 3$ and passes through the point $(4, 5)$.

1. Parallel lines have the same slope. The given line $y = 2x + 3$ is in slope-intercept form $y = mx + b$, where the slope $m = 2$.

2. The equation of a line with slope $m = 2$ passing through $(4, 5)$ can be written using the point-slope form: $y - y_1 = m(x - x_1)$ Substituting $m = 2$, $x_1 = 4$, and $y_1 = 5$: $y - 5 = 2(x - 4)$ Expanding: $y - 5 = 2x - 8 \quad \Rightarrow \quad y = 2x - 3$ Equation: $y = 2x - 3$

b) Is parallel to the line defined by $y = x + 1$ and passes through the point $(1, 3)$.

1. The given line has a slope of $m = 1$ (since $y = x + 1$ means $m = 1$).

2. Using the point-slope form with slope $m = 1$ and point $(1, 3)$: $y - 3 = 1(x - 1)$ Expanding: $y - 3 = x - 1 \quad \Rightarrow \quad y = x + 2$ Equation: $y = x + 2$

c) Is perpendicular to the line defined by $y = -\frac{1}{2}x + 2$ and passes through the point $(3, -5)$.

1. Perpendicular lines have slopes that are negative reciprocals. The slope of the given line is $m = -\frac{1}{2}$, so the slope of the perpendicular line will be $m = 2$ (negative reciprocal of $-\frac{1}{2}$).

2. Using the point-slope form with slope $m = 2$ and point $(3, -5)$: $y - (-5) = 2(x - 3) \quad \Rightarrow \quad y + 5 = 2(x - 3)$ Expanding: $y + 5 = 2x - 6 \quad \Rightarrow \quad y = 2x - 11$ Equation: $y = 2x - 11$

d) Is perpendicular to the line defined by $y = \frac{1}{2}x - 5$ and passes through the point $(-2, -6)$.

1. The slope of the given line is $m = \frac{1}{2}$, so the slope of the perpendicular line will be $m = -2$ (negative reciprocal of $\frac{1}{2}$).

2. Using the point-slope form with slope $m = -2$ and point $(-2, -6)$: $y - (-6) = -2(x - (-2)) \quad \Rightarrow \quad y + 6 = -2(x + 2)$ Expanding: $y + 6 = -2x - 4 \quad \Rightarrow \quad y = -2x - 10$ Equation: $y = -2x - 10$

Recap of Equations:

• a) $y = 2x - 3$
• b) $y = x + 2$
• c) $y = 2x - 11$
• d) $y = -2x - 10$

Would you like further details on any of these parts, or have any other questions?

Related Questions:

1. How do you find the slope of a line given two points?
2. What is the general form of the equation of a line?
3. How do you determine if two lines are perpendicular?
4. What is the point-slope form of a line, and when is it useful?
5. How do you convert from point-slope form to slope-intercept form?

Tip:

When writing equations for parallel lines, remember that only the intercept changes, not the slope.