Question

Which equation represents the line with slope seven over three that passes through the points for, -7

Answer 1

To find the equation of a line with a given slope that passes through a specific point, you can use the point-slope form of the equation of a line, which is:

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

where \(m\) is the slope, \((x_1, y_1)\) is the point the line passes through, and \((x, y)\) are the coordinates of any point on the line.

In your case, the slope \(m = \frac{7}{3}\) and the point is \((4, -7)\).

Substituting these values into the formula, we have:

\[
y - (-7) = \frac{7}{3}(x - 4)
\]

Simplifying this gives:

\[
y + 7 = \frac{7}{3}(x - 4)
\]

Now, distribute \(\frac{7}{3}\) on the right side:

\[
y + 7 = \frac{7}{3}x - \frac{28}{3}
\]

Next, isolate \(y\) by subtracting 7 from both sides:

\[
y = \frac{7}{3}x - \frac{28}{3} - 7
\]

To combine the constant terms, convert 7 into a fraction:

\[
7 = \frac{21}{3}
\]

Now, the equation becomes:

\[
y = \frac{7}{3}x - \frac{28}{3} - \frac{21}{3}
\]

Combine the constant terms:

\[
y = \frac{7}{3}x - \frac{49}{3}
\]

Thus, the equation of the line in slope-intercept form is:

\[
y = \frac{7}{3}x - \frac{49}{3}
\]

Alternatively, in standard form, you could multiply through by 3 to eliminate the fraction:

\[
3y = 7x - 49
\]
Rearranging gives:

\[
7x - 3y = 49
\]

So either of these forms can represent the line you've described.