Question

Use the coordinates of the labeled point to find a point-slope equation of the line.
• (3, -4)
• y +4= 3(x-3)
• y+4=-3(x- 3)
• y-4=-3(x+ 3)
• y-4=3(x+3)

Answers

GPT-4o mini
To create a point-slope equation of a line, we start from the standard point-slope form, which is:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) are the coordinates of a point on the line, and \(m\) is the slope.

In this case, we are given the point (3, -4). So, \(x_1 = 3\) and \(y_1 = -4\).

To find the slope \(m\), we can look at what is typically used for slopes in various equations provided. Given the options and aiming for a valid line from the point (3, -4):

1. **Option 1:** \(y + 4 = 3(x - 3)\) - This translates to a slope of 3.
2. **Option 2:** \(y + 4 = -3(x - 3)\) - This would have a slope of -3.
3. **Option 3:** \(y - 4 = -3(x + 3)\) - In this, the y-intercept would imply it's not through the point (3, -4) correctly.
4. **Option 4:** \(y - 4 = 3(x + 3)\) - Similar to the last, it doesn't address the correct point either.

To find the correct equation using point-slope form, the appropriate slope needs to be defined. Given options 1 and 2 implies either a positive or a negative slope.

Using point (3, -4) and testing option 1:

\[
y + 4 = 3(x - 3)
\]

Expanding this:

\[
y + 4 = 3x - 9
\]
\[
y = 3x - 9 - 4
\]
\[
y = 3x - 13
\]

This is a valid equation starting from the point (3, -4).

Thus, the correct point-slope equation that corresponds to the coordinates (3, -4) is:

\[
y + 4 = 3(x - 3)
\]

So, the finalized answer should be: **y + 4 = 3(x - 3)**.