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Speed is how fast something is moving, measured in distance per unit of time. For example, Andrei ran at 7.7 meters per second. This means every second, he covers 7.7 meters. Similarly, his brother runs at 6.5 meters per second.
Velocity is speed with direction, but in this race, we are concerned only with speed, as both are running in the same direction.
Understanding speed helps us to predict how long it takes to complete an activity, like a race. The relation between distance, speed, and time is given by the formula: \[\text{distance} = \text{speed} \times \text{time}\]This formula helps in figuring out one of the three variables if the other two are known. In our case, the distance for the race is known, and so is the speed. So we can calculate the time.

Linear equations are equations that create a straight line when graphed. They have one or more variables but the highest power of the variable is always one.
In this exercise, the formula for speed, \[ \text{distance} = \text{speed} \times \text{time} \] can be rewritten to determine time by manipulating it into an equation like \[ t = \frac{\text{distance}}{\text{speed}} \]. This results in a simple linear equation because it has only one variable (time) and all operations involved are addition, subtraction, multiplication, or division.
In this race, we have two linear equations:

• For Andrei: \( 50 = 7.7 \times t_a \)
• For the brother: \( 45 = 6.5 \times t_b \)

By solving these equations, we find out how long it takes each of them to finish the race. Linear equations are very useful for solving such real-world problems as they allow for direct computation and understanding of relationships between variables.

Having a structured approach in solving any problem can ease the process significantly. This problem was tackled using key problem-solving steps.
First, understand the problem - this is where you gather all information and clarify what needs to be solved. In this problem, we know both competitors' speeds, starting positions, and the total race distance.
Next, formulate equations - by using the distance formula, set up equations for Andrei and his brother relating their speeds and the total distance.
Once equations are formulated, solve them - this involves basic algebra to isolate the variable of interest. For solving including multiplication or division, like dividing by the speed to find time.
Finally, analyze the results - determine who the winner is by comparing times, and calculate additional info like how much one was ahead at the finish. Thus, following step-by-step instructions makes complex problems seem much simpler and more manageable.