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Part 1
An automobile gets 22.3 miles per gallon​ (mpg) in city driving and30 mpg in highway driving. The car is driven 475 mi on a full tank of 17.4 gal of gasoline. How many miles were driven in the city and how many were driven on the​ highway?

To solve this problem, we'll let $ x $ represent the number of miles driven in the city and $ y $ represent the number of miles driven on the highway.

We know two key facts:
1. The total distance driven is 475 miles.
2. The total amount of gasoline used is 17.4 gallons.

We can create two equations based on these facts:

### Equation 1: Total Distance
\[
x + y = 475
\]

### Equation 2: Total Gasoline Consumption
Gasoline used in the city driving:
\[
\frac{x}{22.3} \quad \text{(miles per gallon for city driving)}
\]
Gasoline used in highway driving:
\[
\frac{y}{30} \quad \text{(miles per gallon for highway driving)}
\]

The total gasoline consumption is:
\[
\frac{x}{22.3} + \frac{y}{30} = 17.4
\]

Now, we have the system of equations:
\[
\begin{align*}
1. & \quad x + y = 475 \\
2. & \quad \frac{x}{22.3} + \frac{y}{30} = 17.4
\end{align*}
\]

Let's solve this system of equations.

### Step 1: Solve the First Equation for One Variable
From Equation 1, solve for $ y $:
\[
y = 475 - x
\]

### Step 2: Substitute $ y $ into the Second Equation
Substitute $ y = 475 - x $ into the second equation:
\[
\frac{x}{22.3} + \frac{475 - x}{30} = 17.4
\]

### Step 3: Solve for $ x $
Multiply through by 669 (the least common multiple of 22.3 and 30) to eliminate the denominators:
\[
669 \times \left(\frac{x}{22.3}\right) + 669 \times \left(\frac{475 - x}{30}\right) = 669 \times 17.4
\]

Simplifying:
\[
30x + 22.3(475 - x) = 11640.6
\]

Expanding and simplifying further:
\[
30x + 10692.5 - 22.3x = 11640.6
\]
\[
7.7x = 11640.6 - 10692.5
\]
\[
7.7x = 948.1
\]
\[
x \approx 123.13 \text{ miles}
\]

### Step 4: Calculate $ y $
Now, substitute $ x = 123.13 $ into the equation $ y = 475 - x $:
\[
y = 475 - 123.13 \approx 351.87 \text{ miles}
\]

### Final Answer
- Miles driven in the city: **123.13 miles**
- Miles driven on the highway: **351.87 miles**

Would you like further details or have any questions?

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Here are five related questions you might find interesting:
1. How much gasoline would the car consume if it drove 600 miles in the city?
2. If the car's highway mileage increased to 35 mpg, how would that affect the gasoline usage for the same trip?
3. What is the total cost of the trip if gasoline costs $3.50 per gallon?
4. How many miles could the car travel if it only drove in the city on a full tank?
5. What would happen to the total miles if the car had 18 gallons of gasoline?

**Tip:** When solving systems of equations, it’s often helpful to solve one equation for a variable and substitute it into the other to reduce the number of variables.

An automobile gets 22.3 miles per gallon (mpg) in city driving and 30 mpg in highway driving. The car is driven 475 mi on a full tank of 17.4 gal of gasoline. How many miles were driven in the city and how many were driven on the highway?

Learn how to solve a real-world problem involving a system of equations to find out how many miles were driven in the city and on the highway. This example involves concepts like algebra, linear equations, and substitution methods, suitable for students in Grades 9-12.

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