for each line, determine whether the slope is positive, negative, zero, or undefined.

Explanation:

Step1: Recall slope - definition

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For a vertical line, $x_2=x_1$, so the denominator is 0. For a horizontal line, $y_2 = y_1$, so the numerator is 0. For a line rising from left - to - right, $y_2>y_1$ and $x_2>x_1$, and for a line falling from left - to - right, $y_2<y_1$ and $x_2>x_1$.

Step2: Analyze Line 1

Line 1 is a vertical line. Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, since $x_2=x_1$ for a vertical line, the denominator is 0. Division by 0 is undefined. So the slope of Line 1 is undefined.

Step3: Analyze Line 2

Line 2 is a line that falls from left - to - right. If we take two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line where $x_2>x_1$, then $y_2<y_1$. So $m=\frac{y_2 - y_1}{x_2 - x_1}<0$. The slope of Line 2 is negative.

Step4: Analyze Line 3

Line 3 is a horizontal line. For two points $(x_1,y_1)$ and $(x_2,y_2)$ on a horizontal line, $y_2 = y_1$. Then $m=\frac{y_2 - y_1}{x_2 - x_1}=0$. The slope of Line 3 is zero.

Step5: Analyze Line 4

Line 4 is a line that falls from left - to - right. If we take two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line with $x_2>x_1$, then $y_2<y_1$. So $m=\frac{y_2 - y_1}{x_2 - x_1}<0$. The slope of Line 4 is negative.

Answer:

Line 1: Undefined Line 2: Negative Line 3: Zero Line 4: Negative