Question

11.2: graphs of two equations
here are two graphs that represent situations you have seen in earlier activities.

1. the first graph represents a = 450 - 20t, which describes the relationship between gallons of water in a tank and time in minutes.

a. where on the graph can we see the 450? where can we see the -20?
b. what do these numbers mean in this situation?

1. the second graph represents 6x + 9y = 75. it describes the relationship between pounds of almonds and figs and the dollar amount clare spent on them.

suppose a classmate says, “i am not sure the graph represents 6x + 9y = 75 because i dont see the 6, 9, or 75 on the graph.” how would you show your classmate that the graph indeed represents this equation?

Explanation:

Step1: Analyze the first - equation's components on the graph

The equation is $a = 450-20t$. The number 450 is the initial value of $a$ (when $t = 0$). On the graph, it is the $y$ - intercept (the value of $a$ when $t=0$). The number - 20 is the slope of the line. It represents the rate of change of the amount of water in the tank with respect to time. A negative slope means the amount of water is decreasing over time.

Step2: Interpret the meaning of the numbers in the context

The 450 means that initially, there are 450 gallons of water in the tank. The - 20 means that the amount of water in the tank is decreasing at a rate of 20 gallons per minute.

Step3: Analyze the second - equation and the graph

For the equation $6x + 9y=75$, we can rewrite it in slope - intercept form $y=-\frac{6}{9}x+\frac{75}{9}=-\frac{2}{3}x+\frac{25}{3}$. To show that the graph represents this equation, we can find the $x$ and $y$ intercepts. When $x = 0$, $y=\frac{75}{9}=\frac{25}{3}\approx8.33$. When $y = 0$, $x=\frac{75}{6}=12.5$. We can also take a few points on the line, substitute their $x$ and $y$ values into the equation $6x + 9y = 75$ and check if the equation holds true.

Answer:

1. a. The 450 is the $y$ - intercept (the value of $a$ when $t = 0$, i.e., the initial amount of water in the tank). The - 20 is the slope of the line, representing the rate of change of the amount of water with respect to time.

b. 450 means there are initially 450 gallons of water in the tank. - 20 means the amount of water is decreasing at a rate of 20 gallons per minute.

1. Rewrite the equation $6x + 9y = 75$ in slope - intercept form $y=-\frac{2}{3}x+\frac{25}{3}$. Find the $x$ and $y$ intercepts ($x$ - intercept: $x = 12.5$ when $y = 0$; $y$ - intercept: $y=\frac{25}{3}\approx8.33$ when $x = 0$) and check if points on the line satisfy the equation $6x + 9y = 75$.