Certainly! Let's find the slope of the line passing through the points [tex]\(P(5, 7)\)[/tex] and [tex]\(Q(-2, -3)\)[/tex] and reduce it to its simplest form.
Given points:
- [tex]\( P = (5, 7) \)[/tex]
- [tex]\( Q = (-2, -3) \)[/tex]
The formula for the slope [tex]\(m\)[/tex] of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates of points [tex]\(P\)[/tex] and [tex]\(Q\)[/tex]:
- For point [tex]\(P\)[/tex], [tex]\((x_1, y_1) = (5, 7)\)[/tex]
- For point [tex]\(Q\)[/tex], [tex]\((x_2, y_2) = (-2, -3)\)[/tex]
Now compute the change in [tex]\(y\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[ \Delta y = y_2 - y_1 = -3 - 7 = -10 \][/tex]
[tex]\[ \Delta x = x_2 - x_1 = -2 - 5 = -7 \][/tex]
So the slope [tex]\(m\)[/tex] is:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{-10}{-7} \][/tex]
The negative signs cancel out in the fraction, yielding:
[tex]\[ m = \frac{10}{7} \][/tex]
In decimal form, this is approximately:
[tex]\[ m = 1.4285714285714286 \][/tex]
Therefore, the slope of the line passing through points [tex]\(P(5, 7)\)[/tex] and [tex]\(Q(-2, -3)\)[/tex] is:
[tex]\[ \text{Slope} = \frac{10}{7} \][/tex]
