Finding Slope - ISEE Upper Level Mathematics Achievement

1. What is the slope of the line through $(-6,1)$ and $(0,-2)$?

   $m=-\frac{1}{2}$. Compute $(-2-1)/(0-(-6))=-3/6=-1/2$.

2. What happens to the slope value if you swap the two points in $m=\frac{y_2-y_1}{x_2-x_1}$?

   Slope stays the same. Swapping points negates both numerator and denominator, preserving the slope value.

3. What is the slope of the line through $(-3,8)$ and $(1,0)$?

   $m=-2$. Slope is $(0-8)/(1-(-3))=-8/4=-2$.

4. What is the slope of the line through $(2,5)$ and $(-1,-4)$?

   $m=3$. Using $(-4-5)/(-1-2)=-9/(-3)=3$.

5. What is the slope of the line through $(4,7)$ and $(0,1)$?

   $m=\frac{3}{2}$. Calculate $(1-7)/(0-4)=-6/(-4)=3/2$.

6. State the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$.

   $m=\frac{y_2-y_1}{x_2-x_1}$. The slope measures the rate of change as the vertical difference divided by the horizontal difference between two points.

7. What is the slope of the line through $(-5,2)$ and $(-1,10)$?

   $m=2$. Formula gives $(10-2)/(-1-(-5))=8/4=2$.

8. What is the slope of the line through $(0,0)$ and $(5,-10)$?

   $m=-2$. Calculate $(-10-0)/(5-0)=-10/5=-2$, showing a downward slope.

9. What is the slope of the line through $(-2,9)$ and $(-8,3)$?

   $m=1$. Compute $(3-9)/(-8-(-2))=-6/(-6)=1$.

10. What is the slope of the line through $(5,12)$ and $(1,4)$?

    $m=2$. Order of points yields $(4-12)/(1-5)=-8/(-4)=2$.

11. What is the slope of the line through $(-4,-6)$ and $(2,3)$?

    $m=\frac{3}{2}$. Using $(3-(-6))/(2-(-4))=9/6=3/2$.

12. What is the slope of the line through $(-1,4)$ and $(3,-2)$?

    $m=-\frac{3}{2}$. Using the formula, $(-2-4)/(3-(-1))=-6/4=-3/2$, indicating a negative slope.

13. What is the slope of the line through $(1,-2)$ and $(4,4)$?

    $m=2$. Compute $(4-(-2))/(4-1)=6/3=2$ using the slope formula.

14. What is the slope of the line through $(2,3)$ and $(6,11)$?

    $m=2$. Apply the slope formula: $(11-3)/(6-2)=8/4=2$, representing rise over run.

15. Find and correct the error: $m=\frac{x_2-x_1}{y_2-y_1}$ for slope between two points.

    Correct: $m=\frac{y_2-y_1}{x_2-x_1}$. The incorrect formula inverts rise over run; the correct one uses change in y over change in x.

16. What is the slope of the line through $(8,-3)$ and $(2,9)$?

    $m=-2$. Calculate $(9-(-3))/(2-8)=12/(-6)=-2$, a negative value.

17. What is the slope of the line through $(-3,5)$ and $(4,5)$?

    $m=0$. Identical y-coordinates yield a numerator of zero, resulting in a slope of zero for horizontal lines.

18. Identify the slope type for a vertical line (in terms of $m$).

    Slope is undefined. Vertical lines have no change in x, causing division by zero and an undefined slope.

19. What is the slope of the line through $(-2,-1)$ and $(2,7)$?

    $m=2$. Using $(7-(-1))/(2-(-2))=8/4=2$, the slope is positive.

20. What is the slope of the line through $(7,2)$ and $(7,9)$?

    Slope is undefined. Identical x-coordinates result in division by zero, making the slope undefined for vertical lines.

21. What is the slope of the line through $(3,1)$ and $(9,0)$?

    $m=-\frac{1}{6}$. Slope is $(0-1)/(9-3)=-1/6$, showing a slight decline.

22. Which condition makes the slope undefined when using $m=\frac{y_2-y_1}{x_2-x_1}$?

    $x_2-x_1=0$. A zero denominator indicates a vertical line, where slope cannot be defined.

23. What is the slope of the line through $(2,-5)$ and $(10,-1)$?

    $m=\frac{1}{2}$. Formula results in $(-1-(-5))/(10-2)=4/8=1/2$.

24. Which condition makes the slope equal to $0$ when using $m=\frac{y_2-y_1}{x_2-x_1}$?

    $y_2-y_1=0$. A zero numerator indicates no vertical change, yielding a horizontal line with zero slope.

25. Identify the slope type for a horizontal line (in terms of $m$).

    $m=0$. Horizontal lines have no change in y, so the slope is zero regardless of x-change.

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