The Following System Of Equations And Accompanying Task Are Presented:1. Equations: $\[ X - 7y = -15 \\] $\[ 3x + Y = 24 \\] $\[ 3x + 5y = 27 \\]2. Talent Show Context: Your School's Talent Show Will
**The Talent Show System of Equations: A Mathematical Puzzle**
The following system of equations and accompanying task are presented:
Equations
{ x - 7y = -15 \}
{ 3x + y = 24 \}
{ 3x + 5y = 27 \}
Talent Show Context
Your school's talent show will feature a variety of acts, including singing, dancing, and magic performances. The talent show committee is responsible for organizing the event and ensuring that everything runs smoothly. As a member of the committee, you have been tasked with solving a system of equations to determine the number of acts that will be performed in each category.
To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.
Step 1: Multiply the First Equation by 3
To eliminate the variable x, we can multiply the first equation by 3.
{ 3(x - 7y) = 3(-15) \}
This simplifies to:
{ 3x - 21y = -45 \}
Step 2: Subtract the Second Equation from the Result
Now, we can subtract the second equation from the result.
{ (3x - 21y) - (3x + y) = -45 - 24 \}
This simplifies to:
{ -22y = -69 \}
Step 3: Solve for y
Now, we can solve for y by dividing both sides of the equation by -22.
{ y = \frac{-69}{-22} \}
This simplifies to:
{ y = 3.136 \}
Step 4: Substitute y into the Second Equation
Now, we can substitute y into the second equation to solve for x.
{ 3x + 3.136 = 24 \}
This simplifies to:
{ 3x = 20.864 \}
Step 5: Solve for x
Now, we can solve for x by dividing both sides of the equation by 3.
{ x = \frac{20.864}{3} \}
This simplifies to:
{ x = 6.955 \}
In conclusion, the system of equations has been solved, and the values of x and y have been determined. The value of x is 6.955, and the value of y is 3.136.
Q: What is the purpose of the talent show system of equations?
A: The purpose of the talent show system of equations is to determine the number of acts that will be performed in each category.
Q: How do you solve a system of equations?
A: There are several methods for solving a system of equations, including the method of substitution and the elimination method.
Q: What is the elimination method?
A: The elimination method is a method for solving a system of equations by eliminating one of the variables.
Q: How do you use the elimination method to solve a system of equations?
A: To use the elimination method, you can multiply one of the equations by a constant and then subtract the other equation from it.
Q: What is the value of x in the talent show system of equations?
A: The value of x in the talent show system of equations is 6.955.
Q: What is the value of y in the talent show system of equations?
A: The value of y in the talent show system of equations is 3.136.
Q: How do you determine the number of acts that will be performed in each category?
A: To determine the number of acts that will be performed in each category, you can use the values of x and y to calculate the number of acts in each category.
Q: What is the significance of the talent show system of equations in real-life scenarios?
A: The talent show system of equations is significant in real-life scenarios because it demonstrates how to solve a system of equations and apply the solution to a real-world problem.
Q: How can you apply the talent show system of equations to other real-life scenarios?
A: You can apply the talent show system of equations to other real-life scenarios by substituting different values for x and y and using the solution to calculate the number of acts in each category.