Math Problem Statement

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?

Solution

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 xx. The goal is to solve for xx 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+1x + 1
    • Distance from Mall to School: x+2x + 2
    • Total distance: (x+1)+(x+2)=2x+3(x + 1) + (x + 2) = 2x + 3
  2. Route passing through the Theater:

    • Distance from Home to Theater: 2x+12x + 1
    • Distance from Theater to School: xx
    • Total distance: (2x+1)+x=3x+1(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+12x + 3 = 3x + 1

Solving for xx:

2x+3=3x+12x + 3 = 3x + 1 Subtract 2x2x from both sides: 3=x+13 = x + 1 Subtract 1 from both sides: x=2x = 2

Part 7b: Finding the total distance from Home to School

Now that we know x=2x = 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(x + 1) + (x + 2) = (2 + 1) + (2 + 2) = 3 + 4 = 7

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


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+12x + 1 instead of x+1x + 1?
  2. What if the distance from Theater to School was 2x2x instead of xx?
  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 xx?
  5. How would the total distance change if xx was negative?

Tip:

Always double-check the distances before solving for xx by writing out each total route explicitly!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

Total distance = Distance from Home to Mall + Distance from Mall to School
Total distance = Distance from Home to Theater + Distance from Theater to School

Theorems

Linear Equations Theorem

Suitable Grade Level

Grades 7-9