Sure, let's determine which equation represents the line that passes through the point [tex]\((3, -7)\)[/tex] and has a slope of [tex]\(\frac{4}{3}\)[/tex].
Step 1: Use the point-slope form of the equation of a line.
The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point, and [tex]\(m\)[/tex] is the slope.
Here, [tex]\((x_1, y_1) = (3, -7)\)[/tex] and [tex]\(m = \frac{4}{3}\)[/tex].
Step 2: Substitute the point and slope into the point-slope form.
[tex]\[ y - (-7) = \frac{4}{3}(x - 3) \][/tex]
Simplifying the left-hand side:
[tex]\[ y + 7 = \frac{4}{3}(x - 3) \][/tex]
Step 3: Convert the point-slope form to slope-intercept form.
[tex]\[ y + 7 = \frac{4}{3}x - \frac{4}{3} \times 3 \][/tex]
[tex]\[ y + 7 = \frac{4}{3}x - 4 \][/tex]
Subtract 7 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{4}{3}x - 4 - 7 \][/tex]
[tex]\[ y = \frac{4}{3}x - 11 \][/tex]
Step 4: Convert the slope-intercept form to standard form [tex]\(Ax + By = C\)[/tex].
First, we clear the fraction by multiplying every term by 3:
[tex]\[ 3y = 4x - 33 \][/tex]
Rearranging to standard form:
[tex]\[ 4x - 3y = 33 \][/tex]
Step 5: Compare the result with the given options.
The correct equation that represents the line passing through the point [tex]\((3, -7)\)[/tex] with a slope of [tex]\(\frac{4}{3}\)[/tex] is:
[tex]\[ 4x - 3y = 33 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{4x - 3y = 33} \][/tex]
