Question

what is the slope of this line? simplify your answer and write it as a proper fraction, improper fraction, or integer.

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify the points

We have two points: \((-5, -3)\) and \((5, 3)\). Let \((x_1, y_1)=(-5, -3)\) and \((x_2, y_2)=(5, 3)\).

Step3: Substitute into the formula

Substitute the values into the slope formula: \( m=\frac{3 - (-3)}{5 - (-5)} \).
Simplify the numerator and denominator: \( 3 - (-3)=3 + 3 = 6 \), \( 5 - (-5)=5 + 5 = 10 \). So \( m=\frac{6}{10} \).

Step4: Simplify the fraction

Simplify \(\frac{6}{10}\) by dividing numerator and denominator by their greatest common divisor, which is 2. So \( \frac{6\div2}{10\div2}=\frac{3}{5} \)? Wait, no, wait: Wait, 6 and 10 have GCD 2? Wait, 6 divided by 2 is 3, 10 divided by 2 is 5? Wait, no, wait, 3 - (-3) is 6, 5 - (-5) is 10. Wait, but also, we can check with the origin. Wait, the line passes through (0,0) as well. Let's use (0,0) and (5,3). Then slope is \( \frac{3 - 0}{5 - 0}=\frac{3}{5} \)? Wait, no, wait, earlier calculation with (-5,-3) and (5,3): \( \frac{3 - (-3)}{5 - (-5)}=\frac{6}{10}=\frac{3}{5} \)? Wait, no, wait, 3 - (-3) is 6, 5 - (-5) is 10, 6/10 reduces to 3/5? Wait, but also, (0,0) and (5,3): slope is 3/5. But wait, let's check again. Wait, (-5,-3) to (5,3): the rise is 3 - (-3)=6, run is 5 - (-5)=10, so 6/10=3/5. Wait, but maybe I made a mistake. Wait, no, let's recalculate. Wait, 3 - (-3) is 6, 5 - (-5) is 10, 6 divided by 10 is 3/5. Wait, but also, the line passes through (0,0), so from (0,0) to (5,3), slope is 3/5. Yes, that's correct.

Wait, no, wait a second. Wait, (-5,-3) and (5,3): the difference in y is 3 - (-3)=6, difference in x is 5 - (-5)=10, so 6/10=3/5. But let's check with (0,0) and (5,3): (3 - 0)/(5 - 0)=3/5. Correct. So the slope is 3/5? Wait, no, wait, maybe I messed up the points. Wait, the point (5,3) and (-5,-3). So y2 - y1 is 3 - (-3)=6, x2 - x1 is 5 - (-5)=10, so 6/10=3/5. Yes.

Answer:

\(\frac{3}{5}\)