Problem 51 - Algebra 2

Question

The caterer at a brunch bought several pounds of chicken salad and several pounds of tuna salad. The chicken salad cost \(\$ 9\) per pound, and the tuna salad cost \(\$ 6\) per pound. He bought a total of 14 pounds of salad and paid a total of \(\$ 111 .\) How much chicken salad did he buy? F. 6 pounds G. 7 pounds H. 8 pounds J. 9 pounds

Step-by-Step Solution

Answer

The caterer purchased 9 pounds of chicken salad.

Step 1: Define Variables

Let the number of pounds of chicken salad be denoted by \(c\). Let the number of pounds of tuna salad be denoted by \(t\). We are asked to find \(c\).

Step 2: Set Up Equations

From the problem, we know two things: the total weight of the salads is 14 pounds, and the total cost is 111 dollars. So, we can write the following equations: 1. \(c + t = 14\) 2. \(9c + 6t = 111\)

Step 3: Solve the System of Equations

First, solve equation 1 for \(t\): \[t = 14 - c\]Now, substitute \(t\) in equation 2:\[9c + 6(14 - c) = 111\]Next, simplify and solve for \(c\): \[9c + 84 - 6c = 111\]\[3c + 84 = 111\]\[3c = 27\]\[c = 9\]

Step 4: Verify the Solution

Substitute \(c = 9\) back into the first equation:\[c + t = 14\]\[9 + t = 14\]\[t = 5\]Check the cost: \[9(9) + 6(5) = 81 + 30 = 111\]This verifies the solution matches the conditions given in the problem.

Key Concepts

Linear Equations

Linear equations are foundational tools in algebra, frequently used to represent relationships with two variables. For this problem, consider the relationship between pounds of salad bought: chicken salad and tuna salad. We used two pieces of information to form our linear equations: the total weight of the salads and the total cost. This gives us:

• The first equation: \( c + t = 14 \) represents the total weight of the salads in pounds. Here, \( c \) stands for pounds of chicken salad, and \( t \) for tuna salad.
• The second equation: \( 9c + 6t = 111 \) stands for the total amount spent on the salads, based on their prices. Chicken salad costs \( \\(9 \) per pound and tuna salad costs \( \\)6 \) per pound.

By setting up these equations, we can solve them to find the quantities of each type of salad. These kinds of systems are often solved using algebraic methods like substitution or elimination. Breaking down problems into linear components often simplifies problem-solving and helps us identify relationships clearly.

Substitution Method

The substitution method is a straightforward technique for solving systems of equations, where we solve one of the equations for one variable, and then substitute that into the other equation. In our exercise, we began by choosing the easier equation, \( c + t = 14 \). Here’s how substitution works step by step:

• First, solve \( c + t = 14 \) for \( t \): \[ t = 14 - c \]
• Next, substitute this expression for \( t \) into the second equation, \( 9c + 6t = 111 \). This leads to: \[ 9c + 6(14 - c) = 111 \]
• Simplify and solve the resulting equation: \[ 9c + 84 - 6c = 111 \] \[ 3c + 84 = 111 \] \[ 3c = 27 \] \[ c = 9 \]

The beauty of this method is its simplicity in isolating one variable at a time. Once you solve for one variable, you can find the other by plugging back into any original equation.

Verification of Solution

Verifying your solution ensures that it meets all the problem's criteria, eliminating any possible errors. Once you've found a possible solution, plug it back into the original equations to confirm it's correct.
In our scenario, we found \( c = 9 \). Now, substitute \( c = 9 \) into the equation \( c + t = 14 \) to find \( t \):

• Substitute to get: \( 9 + t = 14 \)
• Solve for \( t \): \[ t = 14 - 9 \] \[ t = 5 \]

Finally, verify by checking the total cost equation:

• \( 9(9) + 6(5) = 81 + 30 = 111 \)

Both the weight condition \( c + t = 14 \) and the cost condition \( 9c + 6t = 111 \) are satisfied, confirming \( c = 9 \) and \( t = 5 \) is indeed the solution. Verification solidifies your confidence in the result, making it an essential step in solving equations.