Let's go through the questions in the image step-by-step to find the answers.

Question 14

Equation of a line passing through points (-2, 4) and (6, 3).

1. First, calculate the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 4}{6 - (-2)} = \frac{-1}{8} = -\frac{1}{8}$

2. Use the point-slope form $y - y_1 = m(x - x_1)$ with point $(-2, 4)$: $y - 4 = -\frac{1}{8}(x + 2)$

3. Expanding and simplifying: $y - 4 = -\frac{1}{8}x - \frac{1}{4}$ $8y - 32 = -x - 2$ $x + 8y - 30 = 0$ Answer: A. $x + 8y - 30 = 0$

Question 15

Gradient of the line with equation $4x + y = 5$.

1. Rewrite in slope-intercept form $y = mx + b$: $y = -4x + 5$ The gradient (slope) $m = -4$.

   Answer: C. Gradient = -4

Question 16

Equation of a line passing through (0, -5) and parallel to $4x + 2y - 8 = 0$.

1. Rewrite $4x + 2y - 8 = 0$ in slope-intercept form: $2y = -4x + 8$ $y = -2x + 4$ Since parallel lines have the same slope, the slope of the new line is also $-2$.

2. Using point-slope form with point $(0, -5)$: $y + 5 = -2(x - 0)$ $y = -2x - 5$

   Answer: B. $y = -2x - 5$

Question 17

Equation of a line passing through (8, -4) and parallel to $3x + y - 6 = 0$.

1. Rewrite $3x + y - 6 = 0$ in slope-intercept form: $y = -3x + 6$ So, the slope of the new line is also $-3$.

2. Using point-slope form with point $(8, -4)$: $y + 4 = -3(x - 8)$ $y = -3x + 24 - 4$ $y = -3x + 20$ Converting this to standard form: $3x + y - 20 = 0$

   Answer: B. $3x + y - 20 = 0$

Question 18

Equation of a line passing through (3, 4) and parallel to the line passing through points $A(-2, -6)$ and $B(8, 14)$.

1. First, find the slope of the line through points $A$ and $B$: $m = \frac{14 - (-6)}{8 - (-2)} = \frac{20}{10} = 2$

2. Use point-slope form with point $(3, 4)$: $y - 4 = 2(x - 3)$ $y = 2x - 6 + 4$ $y = 2x - 2$ Converting to standard form: $2x - y - 2 = 0$

   Answer: C. $2x - y - 2 = 0$

Would you like more details or have any questions?

Related Questions

1. How do you find the equation of a line passing through two points?
2. What is the point-slope form of a line, and how is it useful?
3. How can you determine if two lines are parallel?
4. How do you convert from slope-intercept form to standard form?
5. What are the key differences between slope-intercept form and standard form of a line?

Tip: When finding equations of parallel lines, remember that parallel lines have identical slopes.