During a back-to-school shopping trip, a group of friends spent $245.86 on 14 shirts and pants. Each shirt cost $11.99. Each pair of pants cost $24.99. How many shirts and pairs of pants did the group buy?
a) Write a system of equations to model the information in the problem.
b) Study the system. Explain, without solving, which method you think would be most efficient for solving the system: substitution, elimination, graphing, or making a table. Explain why the other methods would be less efficient.
c) How could you simplify the numbers used in this system to simplify the system? Does this new system change your answers to part (b)? Explain.
a) Let x be the number of shirts and y be the number of pants.
We can create the following system of equations:
x + y = 14 (equation 1)
11.99x + 24.99y = 245.86 (equation 2)
b) To solve this system, substitution or elimination can be used. Graphing would be less efficient because it would be difficult to accurately graph the equations with decimal coefficients. Making a table would also be less efficient because it would require testing multiple values to find the solution.
c) To simplify the numbers used in the system, we can multiply both sides of equation 2 by 100 to remove the decimal coefficients. This would give us the following system:
x + y = 14 (equation 1)
1199x + 2499y = 24586 (equation 2)
This new system does not change the recommended method for solving (substitution or elimination) because we are only simplifying the coefficients, not changing the nature of the system.
a) Let's use the following variables to represent the unknowns:
Let x be the number of shirts the group bought.
Let y be the number of pants the group bought.
The given information can be translated into the following equations:
Equation 1: x + y = 14 (since the group bought a total of 14 shirts and pants)
Equation 2: 11.99x + 24.99y = 245.86 (since the total amount spent is $245.86)
b) Based on the given system of equations, we have two options for solving it: substitution or elimination.
1. Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation.
2. Elimination: This method involves adding or subtracting the equations together in a way that will eliminate one variable when they are added or subtracted.
Graphing and making a table would be less efficient for this system because it involves decimals (prices) and might not give us accurate solutions.
c) Since the numbers involved in the system have decimals, it would be beneficial to simplify them if possible. However, in this case, the numbers cannot be simplified further because they are already given in decimal form.
Therefore, simplifying the numbers used in the system does not change the answers to part (b).
a) Let's define two variables to represent the number of shirts and pants bought:
Let x be the number of shirts.
Let y be the number of pants.
From the given information, we can set up the following system of equations:
Equation 1: 11.99x + 24.99y = 245.86 (represents the total cost of shirts and pants)
Equation 2: x + y = 14 (represents the total number of shirts and pants)
b) To determine the most efficient method for solving the system, we need to consider the form and simplicity of the equations. Both equations are already in standard form, which is useful for substitution and elimination methods. Graphing and making a table may not be as efficient due to the decimal numbers involved.
Substitution can be an efficient method if one equation is solved for one variable. However, neither equation is currently solved for a variable, so it might not be the most efficient method in this case.
Elimination can also be efficient if one equation can be easily manipulated to eliminate one variable when summed or subtracted with the other equation. The coefficients of the x and y terms in Equation 1 are not simple multiples of the coefficients in Equation 2, making elimination less efficient in this case.
Based on these considerations, substitution might be the most efficient method for solving this system of equations.
c) Simplifying the numbers used in the system can make the calculations easier. In this case, we can multiply both sides of Equation 1 by 100 to get rid of the decimals:
1199x + 2499y = 24586
This modification does not change the form of the equations, so the most efficient method for solving the system remains the same, which is substitution.