29. a basement filled with water from a rainstorm and is draining at a constant rate.
| minutes (x) | liters remaining in basement (y) |
| --- | --- |
| 30 | 1080 |
| 40 | 940 |
| 50 | 780 |

a. write an equation to represent the number of liters in the basement (y) based on the number of minutes the water drains
b. what is the slope?

c. what does the slope represent in the context of the problem? be sure to include units.

d. what is the y - intercept? write as an ordered pair.

e. what does the y - intercept represent in the context of the problem? be sure to include units.

f. what is the x - intercept? write as an ordered pair.

g. what does the x - intercept represent in the context of the problem?

h. how long will it take for there to be only 500 liters remaining in the basement?

30. find the missing coordinate of the line with a slope of $\frac{1}{4}$ and goes through points $(7,10)$ and $(- 8,y)$.

31. find the missing coordinate of the line that has a slope of 5 and goes through points $(- 4,4)$ and $(x,-6)$.

Question

29. a basement filled with water from a rainstorm and is draining at a constant rate.
| minutes (x) | liters remaining in basement (y) |
| --- | --- |
| 30 | 1080 |
| 40 | 940 |
| 50 | 780 |

a. write an equation to represent the number of liters in the basement (y) based on the number of minutes the water drains
b. what is the slope?

c. what does the slope represent in the context of the problem? be sure to include units.

d. what is the y - intercept? write as an ordered pair.

e. what does the y - intercept represent in the context of the problem? be sure to include units.

f. what is the x - intercept? write as an ordered pair.

g. what does the x - intercept represent in the context of the problem?

h. how long will it take for there to be only 500 liters remaining in the basement?

30. find the missing coordinate of the line with a slope of $\frac{1}{4}$ and goes through points $(7,10)$ and $(- 8,y)$.

31. find the missing coordinate of the line that has a slope of 5 and goes through points $(- 4,4)$ and $(x,-6)$.

Accepted Answer

### Part 29 (a)
# Explanation:
## Step 1: Calculate the slope
We use two points $(x_1,y_1)=(20,1060)$ and $(x_2,y_2)=(40,840)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. So $m=\frac{840 - 1060}{40 - 20}=\frac{- 220}{20}=- 11$.
## Step 2: Use point - slope form to find the equation
The point - slope form is $y - y_1=m(x - x_1)$. Using the point $(20,1060)$ and $m=-11$, we have $y - 1060=-11(x - 20)$.
Expand the right - hand side: $y-1060=-11x + 220$.
Then, add 1060 to both sides: $y=-11x+220 + 1060$, so $y=-11x + 1280$.
# Answer: $y=-11x + 1280$

### Part 29 (b)
# Explanation:
From the equation of the line $y = mx + b$ (slope - intercept form), where $m$ is the slope. In the equation $y=-11x + 1280$, the slope $m=-11$. We can also calculate it using two points $(x_1,y_1)=(20,1060)$ and $(x_2,y_2)=(40,840)$ as $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{840 - 1060}{40 - 20}=\frac{-220}{20}=-11$.
# Answer: $-11$

### Part 29 (c)
# Explanation:
The slope of a linear equation in the form $y = mx + b$ (where $y$ is the amount of water in liters and $x$ is the time in minutes) represents the rate of change of $y$ with respect to $x$. Since $m=-11$, it means that the number of liters of water in the basement is decreasing at a rate of 11 liters per minute.
# Answer: The slope represents the rate at which the water is draining from the basement, and it is - 11 liters per minute (the negative sign indicates a decrease in the amount of water as time increases).

### Part 29 (d)
# Explanation:
The $y$ - intercept of the line $y=mx + b$ is the point where $x = 0$. In the equation $y=-11x + 1280$, when $x = 0$, $y=1280$. So the $y$ - intercept as an ordered pair is $(0,1280)$.
# Answer: $(0,1280)$

### Part 29 (e)
# Explanation:
The $y$ - intercept occurs when $x = 0$ ( $x$ represents the number of minutes the water has been draining). So when $x = 0$ (at the start, before any water has drained), the number of liters of water in the basement is 1280 liters.
# Answer: The $y$ - intercept represents the initial amount of water in the basement before any water has drained, and it is 1280 liters.

### Part 29 (f)
# Explanation:
The $x$ - intercept is the point where $y = 0$. Set $y = 0$ in the equation $y=-11x + 1280$.
We have the equation $0=-11x + 1280$.
Add $11x$ to both sides: $11x=1280$.
Then, $x=\frac{1280}{11}\approx116.36$. So the $x$ - intercept as an ordered pair is $(\frac{1280}{11},0)$ or approximately $(116.36,0)$.
# Answer: $(\frac{1280}{11},0)$ (or approximately $(116.36,0)$)

### Part 29 (g)
# Explanation:
The $x$ - intercept is the value of $x$ (time in minutes) when $y = 0$ (the amount of water in liters is 0). So it represents the number of minutes it takes for the basement to be completely drained of water.
# Answer: The $x$ - intercept represents the number of minutes it takes for the basement to have 0 liters of water (i.e., the time when the basement is completely drained).

### Part 29 (h)
# Explanation:
We know that $y = 500$ (the number of liters remaining) and the equation of the line is $y=-11x + 1280$.
Set $y = 500$ in the equation: $500=-11x + 1280$.
Subtract 1280 from both sides: $500-1280=-11x$.
$-780=-11x$.
Divide both sides by $-11$: $x=\frac{780}{11}\approx70.91$ minutes.
# Answer: $\frac{780}{11}$ minutes (or approximately 70.91 minutes)

### Part 30
# Explanation:
## Step 1: Recall the slope formula
The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. We know that $m = \frac{1}{4}$, $(x_1,y_1)=(7,10)$ and $(x_2,y_2)=(-8,y)$.
So $\frac{1}{4}=\frac{y - 10}{-8 - 7}$.
## Step 2: Solve for $y$
First, simplify the denominator: $-8 - 7=-15$. So the equation becomes $\frac{1}{4}=\frac{y - 10}{-15}$.
Cross - multiply: $y - 10=\frac{-15}{4}$.
Add 10 to both sides: $y=10-\frac{15}{4}=\frac{40 - 15}{4}=\frac{25}{4}=6.25$.
# Answer: $y=\frac{25}{4}$ (or $y = 6.25$)

### Part 31
# Explanation:
## Step 1: Use the slope formula
The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. We know that $m = 5$, $(x_1,y_1)=(-4,4)$ and $(x_2,y_2)=(x,-6)$.
So $5=\frac{-6 - 4}{x-(-4)}$.
## Step 2: Solve for $x$
Simplify the numerator: $-6 - 4=-10$. The equation becomes $5=\frac{-10}{x + 4}$.
Cross - multiply: $5(x + 4)=-10$.
Expand the left - hand side: $5x+20=-10$.
Subtract 20 from both sides: $5x=-10 - 20=-30$.
Divide both sides by 5: $x=-6$.
# Answer: $x=-6$

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QUESTION IMAGE

Question

Explanation:

Part 29 (a)

Step 1: Calculate the slope

Step 2: Use point - slope form to find the equation

Answer:

Part 29 (b)