b. Use the equations to determine where the two lines intersect, Find a value for both variables. Label $$ \begin{array}{l} \text { this point on the graph. } \ \begin{array}{l} 86000=8000 t=102000-12000 t \ 86000=-1200000+8000 \end{array} \ -16000=-4000 t \quad t=\frac{16000}{1000}=4 \end{array} $$

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UpStudy AI · Accepted Answer

It looks like you're working to find the intersection point of two lines by solving a system of equations. Let's go through the steps to determine the correct values for both variables and accurately label the intersection point on the graph.

## Understanding the Problem

Assuming you have two linear equations representing two lines:

1. **Equation 1:** $$ y = 8000t $$
2. **Equation 2:** $$ y = 102000 - 12000t $$

Your goal is to find the values of $$ t $$ and $$ y $$ where these two lines intersect, meaning both equations are equal at that point.

## Step-by-Step Solution

### 1. Set the Equations Equal to Each Other

Since both equations equal $$ y $$, set them equal to each other to find the value of $$ t $$:

$$
8000t = 102000 - 12000t
$$

### 2. Combine Like Terms

Add $$ 12000t $$ to both sides to gather all $$ t $$ terms on one side:

$$
8000t + 12000t = 102000
$$

$$
20000t = 102000
$$

### 3. Solve for $$ t $$

Divide both sides by $$ 20000 $$ to isolate $$ t $$:

$$
t = \frac{102000}{20000} = 5.1
$$

### 4. Substitute $$ t $$ Back into One of the Original Equations to Find $$ y $$

You can use either equation; we'll use **Equation 1** for simplicity:

$$
y = 8000t = 8000 \times 5.1 = 40800
$$

### 5. Determine the Intersection Point

The lines intersect at the point $$ (t, y) = (5.1, 40800) $$.

## Corrected Calculation

Your original steps seem to have some discrepancies. Here's a corrected version based on the standard method:

$$
\begin{align*}
8000t &= 102000 - 12000t \
8000t + 12000t &= 102000 \
20000t &= 102000 \
t &= \frac{102000}{20000} \
t &= 5.1 \
y &= 8000 \times 5.1 = 40800 \
\end{align*}
$$

So, the correct values are:
- $$ t = 5.1 $$
- $$ y = 40,\!800 $$

## Graphical Representation

To **label the intersection point on the graph**, plot both lines using their respective equations and mark the point where they cross.

1. **Plotting $$ y = 8000t $$:**
   - For $$ t = 0 $$, $$ y = 0 $$
   - For $$ t = 5.1 $$, $$ y = 40,\!800 $$

2. **Plotting $$ y = 102000 - 12000t $$:**
   - For $$ t = 0 $$, $$ y = 102,\!000 $$
   - For $$ t = 5.1 $$, $$ y = 40,\!800 $$

The lines will intersect at $$ (5.1, 40,\!800) $$. Here's a simple sketch:

```
|
|                  *
|                / 
|              /
|            /
|          /
|        /
|      /
|    /
|  /
|/________________________
```

- The asterisk (*) represents the intersection point $$ (5.1, 40,\!800) $$.

## Conclusion

The two lines intersect at the point $$ (t, y) = (5.1, 40,\!800) $$. Ensure that when setting up and solving your equations, all terms are correctly accounted for to avoid discrepancies in your final answer.
The two lines intersect at the point $$ (5.1, 40,\!800) $$.

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