91Ó°ÊÓ

Algebraic equations are the foundation of many mathematical problems. These equations include variables, constants, and arithmetic operations. In the given problem, we encounter two algebraic equations representing the speeds of two cars. An algebraic equation can be as simple as a single variable equating two numbers (e.g., \(x = 2\)) or more complex involving multiple variables and operations (e.g., \(6r_1 + 6r_2 = 600\)).

In our problem, by defining the speed of one car as \(r_1\) and knowing the second car's speed is 30 mph slower (\(r_2 = r_1 - 30\)), we set up relationships between the variables. Solving algebraic equations often involves:

• Substituting known values
• Simplifying terms
• Isolating the variable

To solve any equation, follow structured steps. We started by defining variables because it helps us translate word problems into mathematical form. Next, we set up the primary equation based on the relationship and distances: \(6r_1 + 6(r_1 - 30) = 600\). By substituting \(r_2 = r_1 - 30\) into the equation:

We simplify the equation into a single variable:
\ \[ 6r_1 + 6r_1 - 180 = 600 \]
Combining like terms, we get:
\ \[ 12r_1 - 180 = 600 \]
Isolate \(r_1\), first by adding 180:
\ \[ 12r_1 = 780 \]
Then, divide by 12:

\ \[ r_1 = 65 \].

Finally, we find the second variable \(r_2\) by substituting back: \ \[ r_2 = 65 - 30 = 35 \]

Each step is crucial to ensure precision. It helps prevent mistakes and confirm the solution matches all conditions in the problem.

Distance-rate-time problems link these three elements:

• Distance (D): How far an object travels
• Rate (R): Speed of the object
• Time (T): Duration of the travel

These problems use the fundamental equation: \ \[ D = R \times T \]
In this problem, distances of two cars traveling towards each other add up. Each distance can be determined by multiplying their speed by travel time (6 hours): \ \[ D = r_1 \times 6 + (r_1 - 30) \times 6 \]
We can create equations to solve for unknowns. By understanding and setting up a system of equations, such problems become manageable and practical for real-life scenarios like travel planning or logistics. These problems also teach us how to break down complex scenarios into simpler steps, ensuring accurate solutions through logical problem-solving.