https://file.aiquickdraw.com/mathbot/file/17270370052614mjk69wy.jpg

Let's begin by solving the equations of the straight lines in the form \(y = mx + c\) (slope-intercept form), based on the problems listed in the image.

### First Set of Problems:

1. **Find the equation of the straight line that:**

   a) **Has a gradient (slope) of 3 and passes through (-3, 3):**

   We use the point-slope form:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \(m = 3\), \((x_1, y_1) = (-3, 3)\).

   Substituting:
   \[
   y - 3 = 3(x + 3)
   \]
   Simplifying:
   \[
   y = 3x + 9 + 3
   \]
   Final equation: 
   \[
   y = 3x + 12
   \]

   b) **Has a gradient of -0.5 and passes through (0, 0):**

   Using the same formula with \(m = -0.5\) and \((x_1, y_1) = (0, 0)\):
   \[
   y - 0 = -0.5(x - 0)
   \]
   Simplifying:
   \[
   y = -0.5x
   \]

   c) **Is parallel to \(2y = 5x + 7\) and passes through (0, -3.5):**

   First, rearrange \(2y = 5x + 7\) into slope-intercept form \(y = mx + c\):
   \[
   y = \frac{5}{2}x + \frac{7}{2}
   \]
   The slope is \(\frac{5}{2}\), and since the lines are parallel, their slopes are equal. Using point (0, -3.5):
   \[
   y + 3.5 = \frac{5}{2}(x - 0)
   \]
   Simplifying:
   \[
   y = \frac{5}{2}x - 3.5
   \]

   d) **Is parallel to \(9x - 5y = -3\) and passes through (5, -3):**

   First, rearrange \(9x - 5y = -3\) to get it in slope-intercept form:
   \[
   -5y = -9x - 3
   \]
   \[
   y = \frac{9}{5}x + \frac{3}{5}
   \]
   The slope is \(\frac{9}{5}\), and using the point (5, -3):
   \[
   y + 3 = \frac{9}{5}(x - 5)
   \]
   Simplifying:
   \[
   y = \frac{9}{5}x - 12
   \]

### Second Set of Problems:

2. **Find the equation of the straight line joining:**

   a) **A is (-2, 1) and B is (-1, 4):**

   The slope \(m\) is given by:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{-1 - (-2)} = \frac{3}{1} = 3
   \]
   Using point (-2, 1):
   \[
   y - 1 = 3(x + 2)
   \]
   Simplifying:
   \[
   y = 3x + 7
   \]

   b) **A is (-3, 4) and B is (6, 1):**

   The slope \(m\) is:
   \[
   m = \frac{1 - 4}{6 - (-3)} = \frac{-3}{9} = -\frac{1}{3}
   \]
   Using point (-3, 4):
   \[
   y - 4 = -\frac{1}{3}(x + 3)
   \]
   Simplifying:
   \[
   y = -\frac{1}{3}x + 3
   \]

   c) **A is (-2, 1) and B is (3, 1):**

   Since the y-coordinates are the same, the line is horizontal:
   \[
   y = 1
   \]

### Third Set of Problems:

3. **A is (-10, 5) and B is (8, -4):**

   a) The slope \(m\) is:
   \[
   m = \frac{-4 - 5}{8 - (-10)} = \frac{-9}{18} = -\frac{1}{2}
   \]
   Using point (-10, 5):
   \[
   y - 5 = -\frac{1}{2}(x + 10)
   \]
   Simplifying:
   \[
   y = -\frac{1}{2}x
   \]

I have solved the initial parts. Would you like further assistance on this?

Find the equations of various straight lines and solve related geometric problems.

Solve straight-line equations with different slopes, including lines passing through given points, lines parallel to others, and more. Learn how to apply point-slope and slope-intercept forms to find the equations. Ideal for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation