What is the equation of the line that passes through the point (4, 2) and is parallel to the line given by the equation 3x + 2y = 6?

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Understanding the Problem

To find the equation of a line that is parallel to another, we first need to identify the slope of the line we are targeting. In this case, we are given the equation of a line in standard form:

3x + 2y = 6

Step 1: Find the Slope of the Given Line

To derive the slope from the equation, we need to convert it into the slope-intercept form, which is:

y = mx + b

Where m represents the slope. Here are the conversion steps:

  1. Start by isolating the y variable.
  2. Reorganize the equation:
    3. 2y = -3x + 6
  3. Divide every term by 2:
    4. y = -1.5x + 3

From this, we can see that the slope (m) of the line is -1.5.

Step 2: Determine the Slope of the Parallel Line

A line parallel to another will have the same slope. Therefore, the slope of our desired line will also be -1.5.

Step 3: Use the Point-Slope Form of the Line Equation

We will use the point-slope form of the line equation to find our desired line, as it’s ideal for creating a line through a specific point:

y – y1 = m(x – x1)

Where:

  • m is the slope,
  • (x1, y1) is the point through which the line passes.

Here, we have:

  • m = -1.5
  • (x1, y1) = (4, 2)

Plugging these values into the point-slope formula yields:

y - 2 = -1.5(x - 4)

Step 4: Simplify to Slope-Intercept Form

Next, we simplify this equation to convert it to the slope-intercept form:

  1. Distributing -1.5:
    2. y - 2 = -1.5x + 6
  2. Now, add 2 to both sides:
    3. y = -1.5x + 8

Conclusion

Thus, the equation of the line that passes through the point (4, 2) and is parallel to the line 3x + 2y = 6 is:

y = -1.5x + 8

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