Here’s how to solve the problems in the image step by step:

2-70: Finding x- and y-intercepts

For the given equations, we find the x- and y-intercepts by substituting 0 for one variable at a time.

(a) $x - 2y = 4$

1. x-intercept: Let $y = 0$. $x - 2(0) = 4 \implies x = 4$ So, the x-intercept is $(4, 0)$.

2. y-intercept: Let $x = 0$. $0 - 2y = 4 \implies y = -2$ So, the y-intercept is $(0, -2)$.

(b) $3x + 6y = 24$

1. x-intercept: Let $y = 0$. $3x + 6(0) = 24 \implies x = 8$ So, the x-intercept is $(8, 0)$.

2. y-intercept: Let $x = 0$. $3(0) + 6y = 24 \implies y = 4$ So, the y-intercept is $(0, 4)$.

2-71: Finding the slope

The formula for slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

(a) $(1, 2)$ and $(4, -1)$

$m = \frac{-1 - 2}{4 - 1} = \frac{-3}{3} = -1$

(b) $(7, 3)$ and $(5, 4)$

$m = \frac{4 - 3}{5 - 7} = \frac{1}{-2} = -\frac{1}{2}$

(c) $(-6, 8)$ and $(-8, 5)$

$m = \frac{5 - 8}{-8 - (-6)} = \frac{-3}{-2} = \frac{3}{2}$

(d) $(55, 67)$ and $(50, 68)$

$m = \frac{68 - 67}{50 - 55} = \frac{1}{-5} = -\frac{1}{5}$

(e) Azizah's mistake

Azizah incorrectly found the slope of $(1, 2)$ and $(4, -1)$ as 1 instead of $-1$. She likely swapped the subtraction of coordinates or made an arithmetic error. To find the slope: $m = \frac{-1 - 2}{4 - 1} = \frac{-3}{3} = -1$ She needs to carefully subtract the coordinates in the proper order, $(y_2 - y_1)$ over $(x_2 - x_1)$.

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Related Questions

1. What is the significance of x- and y-intercepts in graphing a line?
2. How does the slope indicate the steepness and direction of a line?
3. Can you convert the given equations into slope-intercept form for graphing?
4. What common errors occur when calculating slope between two points?
5. How are intercepts used to predict the behavior of linear equations?

Tip

Always double-check calculations, especially when subtracting negative numbers in slope computations!