https://file.aiquickdraw.com/mathbot/file/1730080969415orynihj8.jpg

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

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### 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. \( x + 8y - 30 = 0 \)**

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### 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**

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### 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. \( y = -2x - 5 \)**

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### 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. \( 3x + y - 20 = 0 \)**

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### 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. \( 2x - y - 2 = 0 \)**

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Would you like more details or have any questions?

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### 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?

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

Find the solutions for questions 14 to 18 on linear equations and slopes.

A step-by-step solution for finding equations of lines through given points and determining parallel lines, suitable for high school students. Topics include calculating slopes, using point-slope and slope-intercept forms, and converting between line forms.

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