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Defining variables at the start of solving a system of equations helps to structure the problem. It allows us to translate the word problem into mathematical language. In our car salesperson example, we denote the number of small cars by S, medium cars by M, and large cars by L. This simple step makes it easier to form equations based on the problem's conditions. When defining your variables, choose symbols that make sense, which will help you keep track of what each symbol represents as you work through the problem.

After defining the variables, the next step involves translating the problem's conditions into equations. These equations will allow us to solve for the unknown values. First, it's essential to set up the equations correctly. From our problem, we know:

• Total cars sold is 24: S + M + L = 24
• She must sell 3 more small cars than medium cars: S = M + 3
• The number of medium cars is equal to the number of large cars: M = L

Once we have our equations, we substitute one equation into another to simplify the system. For example, we substitute S with (M + 3) and L with M in the total cars equation:
ewline ewline (M + 3) + M + M = 24
Simplifying the equations step-by-step will bring us closer to finding the exact values of each variable.

Balancing equations is a crucial step as it ensures that both sides of an equation are equal, which is fundamental in solving systems of equations. Once we've substituted and set up our equations as
ewline (M + 3) + M + M = 24,
ewline it's time to balance and simplify. Combine like terms:

3M + 3 = 24.
Next, isolate the variable by subtracting 3 from both sides:
ewline 3M = 21.
ewline Finally, divide by 3 to solve for M:
ewline M = 7.
ewline Now with M known, we use it to find S and L. Since S = M + 3:
S = 7 + 3 = 10.
And since M = L: L = 7.
ewline Verifying these values ensures they meet all the original conditions. Balancing your equations at each step is essential for making sure your solution is accurate and satisfies all parts of the problem.