Question

1. a cyclist is riding along a trail. her distance travelled can be represented by the equation \\(d = 20t + 15\\). a second cyclists distance on the same trail is given by the equation \\(d = 25t + 5\\). in both equations, \\(d\\) is distance in km and \\(t\\) is time in hours.

a. graph the system to determine the solution.
b. explain what the solution represents in the context of the question.

1. create and solve graphically a system of linear equations. the admission fee to an interpretive center is \\$5 for a student and \\$9 for an adult. one day, 32 people entered the center and a total of \\$180 in admission fees was collected.

Explanation:

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<reasoning>

Solve the system of equations for Question 3

\[

$$\begin{aligned} &d = 20t + 15 \\ &d = 25t + 5 \\ &20t + 15 = 25t + 5 \\ &5t = 10 \implies t = 2 \\ &d = 20(2) + 15 = 55 \end{aligned}$$

\]

Formulate the system of equations for Question 4

\[

$$\begin{aligned} &x + y = 32 \\ &5x + 9y = 180 \\ &y = 32 - x \\ &5x + 9(32 - x) = 180 \\ &-4x + 288 = 180 \implies 4x = 108 \implies x = 27 \\ &y = 32 - 27 = 5 \end{aligned}$$

\]
</reasoning>

<answer>

Question 3

a. The system of equations is:
\[

$$\begin{aligned} d &= 20t + 15 \\ d &= 25t + 5 \end{aligned}$$

\]
The intersection point of the two lines on the graph is \((2, 55)\).

b. The solution represents that after \(2\) hours, both cyclists have travelled the exact same distance of \(55\text{ km}\).

Question 4

The system of equations representing the situation is:
\[

$$\begin{aligned} x + y &= 32 \\ 5x + 9y &= 180 \end{aligned}$$

\]
where \(x\) is the number of students and \(y\) is the number of adults.

Solving the system yields:
\[

$$\begin{aligned} x &= 27\text{ students} \\ y &= 5\text{ adults} \end{aligned}$$

\]
The graphical solution is the intersection point \((27, 5)\).
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "20*x + 15", "latex": "20t + 15"}, 0, 5],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "d = 20t + 15",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "25*x + 5", "latex": "25t + 5"}, 0, 5],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"name": "d = 25t + 5",
"withLabel": true
}
},
{
"type": "point",
"params": [[2, 55]],
"properties": {
"name": "(2, 55)",
"size": 4,
"color": "#583C87",
"withLabel": true
}
}
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"timestamps": [0.5, 1.0]
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</plot>

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Answer:

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<reasoning>

Solve the system of equations for Question 3

\[

$$\begin{aligned} &d = 20t + 15 \\ &d = 25t + 5 \\ &20t + 15 = 25t + 5 \\ &5t = 10 \implies t = 2 \\ &d = 20(2) + 15 = 55 \end{aligned}$$

\]

Formulate the system of equations for Question 4

\[

$$\begin{aligned} &x + y = 32 \\ &5x + 9y = 180 \\ &y = 32 - x \\ &5x + 9(32 - x) = 180 \\ &-4x + 288 = 180 \implies 4x = 108 \implies x = 27 \\ &y = 32 - 27 = 5 \end{aligned}$$

\]
</reasoning>

<answer>

Question 3

a. The system of equations is:
\[

$$\begin{aligned} d &= 20t + 15 \\ d &= 25t + 5 \end{aligned}$$

\]
The intersection point of the two lines on the graph is \((2, 55)\).

b. The solution represents that after \(2\) hours, both cyclists have travelled the exact same distance of \(55\text{ km}\).

Question 4

The system of equations representing the situation is:
\[

$$\begin{aligned} x + y &= 32 \\ 5x + 9y &= 180 \end{aligned}$$

\]
where \(x\) is the number of students and \(y\) is the number of adults.

Solving the system yields:
\[

$$\begin{aligned} x &= 27\text{ students} \\ y &= 5\text{ adults} \end{aligned}$$

\]
The graphical solution is the intersection point \((27, 5)\).
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "20*x + 15", "latex": "20t + 15"}, 0, 5],
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"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "d = 20t + 15",
"withLabel": true
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{
"type": "functiongraph",
"params": [{"js": "25*x + 5", "latex": "25t + 5"}, 0, 5],
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"name": "d = 25t + 5",
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"params": [[2, 55]],
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"name": "(2, 55)",
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