https://file.aiquickdraw.com/mathbot/file/1727406472722q6dxf1q6.jpg

The problem shown in the image presents a situation where different routes from "Home" to "School" involve different points of reference: the "Mall" and the "Theater." The distances between each point are expressed in terms of \(x\). The goal is to solve for \(x\) using the given information that the total distance passing through the Mall equals the total distance passing through the Theater.

### Part 7a: Setting up the equation
1. Route passing through the Mall:
   - Distance from Home to Mall: \(x + 1\)
   - Distance from Mall to School: \(x + 2\)
   - Total distance: \((x + 1) + (x + 2) = 2x + 3\)

2. Route passing through the Theater:
   - Distance from Home to Theater: \(2x + 1\)
   - Distance from Theater to School: \(x\)
   - Total distance: \((2x + 1) + x = 3x + 1\)

Since the problem states that these two routes are the same, you can set the total distances equal to each other:

\[
2x + 3 = 3x + 1
\]

### Solving for \(x\):
\[
2x + 3 = 3x + 1
\]
Subtract \(2x\) from both sides:
\[
3 = x + 1
\]
Subtract 1 from both sides:
\[
x = 2
\]

### Part 7b: Finding the total distance from Home to School
Now that we know \(x = 2\), we can plug this value back into either route to find the total distance. Let's use the route through the Mall:

\[
(x + 1) + (x + 2) = (2 + 1) + (2 + 2) = 3 + 4 = 7
\]

Therefore, the total distance from Home to School is **7 miles**.

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Would you like any more details or have any questions?

### Related Questions:
1. How would the equation change if the distance from Home to Mall was \(2x + 1\) instead of \(x + 1\)?
2. What if the distance from Theater to School was \(2x\) instead of \(x\)?
3. How do you solve systems of linear equations with more than two variables?
4. How can you verify that both routes still give the same total distance for different values of \(x\)?
5. How would the total distance change if \(x\) was negative?

### Tip:
Always double-check the distances before solving for \(x\) by writing out each total route explicitly!

From your home, the route to the school that passes the mall is the same as the route to the school that passes the theater. Use what you know about the routes to school to write an equation and solve for x. How many miles is it from your home to the school?

Learn how to solve a real-life algebra problem involving routes and distances. Use linear equations to find the value of x and calculate the total distance from home to school. Suitable for students in Grades 7-9.

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