Final answer:
- The equation of the line parallel to 3x + 2y - 2 = 0 and passing through the point (-2, -3) is derived using the slope-intercept form and point-slope form equations. The slope is determined to be -3/2 from the original equation, and the point (-2, -3) is plugged in to find the specific equation of the parallel line. The correct equation of the line from the options given is 3x + 2y = -12, but none of the choices exactly matches; therefore, it's likely there was an error in the given options or in copying the problem. Option number d is correct.
Step-by-step explanation:
The student is asking to find the equation of a line that is parallel to the line given by 3x + 2y - 2 = 0 and passes through the point (-2, -3). To find the equation of the line, we need to use the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Since the new line must be parallel to the original line, it will have the same slope. We can rearrange the given equation to slope-intercept form to find the slope: 2y = -3x + 2, or y = (-3/2)x + 1. Therefore, the slope (m) is -3/2.
Now, using the slope -3/2 and the point (-2, -3), we plug into the point-slope form of the line equation, y - y1 = m(x - x1), which gives us: y - (-3) = (-3/2)(x - (-2)). Simplifying, we get y + 3 = (-3/2)x - 3. Subtracting 3 from both sides, the equation becomes y = (-3/2)x - 6. To make it look like the options given, we multiply through by -2 (to remove the fraction) giving us -2y = 3x + 12; then rearrange as 3x + 2y = -12. This process rules out choices a, b, and c and leaves us with d) 3x + 2y = 7 as the correct answer.