Question

1 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 on the graph can we see the -20?
b. what do these numbers mean in this situation?
2 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 that 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 equation \(a = 450-20t\)

The equation is in the form \(y = mx + b\) (here \(a\) is like \(y\) and \(t\) is like \(x\)). In the \(y=mx + b\) form, \(b\) is the \(y\) - intercept. When \(t = 0\), \(a=450-20\times0=450\). So, the \(y\) - intercept of the first graph (the value of \(a\) when \(t = 0\)) shows the \(450\).
The slope \(m=-20\). The slope \(m=\frac{\Delta y}{\Delta x}=\frac{\Delta a}{\Delta t}\). The slope represents the rate of change.

Step2: Interpret the meaning of \(450\) and \(- 20\)

For \(a = 450-20t\), when \(t = 0\) (at the start, time \(t = 0\) minutes), the amount of water in the tank \(a = 450\) gallons. So, \(450\) is the initial amount of water in the tank.
The coefficient \(-20\) in \(a = 450-20t\) is the slope. Since \(a\) is in gallons and \(t\) is in minutes, the slope \(\frac{\Delta a}{\Delta t}=-20\) means that the amount of water in the tank is decreasing at a rate of \(20\) gallons per minute.

Step3: Analyze the equation \(6x + 9y=75\)

We can rewrite the equation \(6x + 9y=75\) in slope - intercept form \(y=mx + b\). First, solve for \(y\):

$$
LATEXBLOCK0
$$

We can also find the \(x\) - intercept and \(y\) - intercept.
For the \(x\) - intercept, set \(y = 0\):

$$
LATEXBLOCK1
$$

For the \(y\) - intercept, set \(x = 0\):

$$
LATEXBLOCK2
$$

The \(x\) - intercept \((\frac{25}{2},0)\) and \(y\) - intercept \((0,\frac{25}{3})\) are related to the coefficients in the equation \(6x + 9y=75\). We can also check points on the line. For example, if \(x = 3\):

$$
LATEXBLOCK3
$$

Answer:

1.
a. The \(450\) is the \(y\) - intercept (the value of \(a\) when \(t = 0\)) of the first graph. The \(-20\) is the slope of the first graph.
b. \(450\) is the initial amount of water (in gallons) in the tank. \(-20\) means the amount of water in the tank 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 \(x\) - intercept \((\frac{25}{2},0)\) (by setting \(y = 0\): \(6x=75\Rightarrow x=\frac{75}{6}=\frac{25}{2}\)) and \(y\) - intercept \((0,\frac{25}{3})\) (by setting \(x = 0\): \(9y=75\Rightarrow y=\frac{75}{9}=\frac{25}{3}\)). Also, we can pick values of \(x\) (e.g., \(x = 3\)) and solve for \(y\) (\(6\times3+9y=75\Rightarrow y=\frac{19}{3}\)) and check if the point \((3,\frac{19}{3})\) lies on the graph.