Problem 88 - Algebra 2

Question

FURNITURE A new sofa, love seat, and coffee table cost \(\$ 2050\) . The sofa costs twice as much as the love seat. The sofa and the coffee table together cost \(\$ 1450 .\) How much does each piece of furniture cost?

Step-by-Step Solution

Answer

The love seat costs $600, the sofa costs $1200, and the coffee table costs $250.

Step 1: Define Variables

Let the cost of the love seat be \( L \), the cost of the sofa be \( S \), and the cost of the coffee table be \( C \). We know from the problem that \( S = 2L \) since the sofa costs twice as much as the love seat.

Step 2: Setup Equations

From the problem, we have two main equations: \( S + L + C = 2050 \) for the total cost and \( S + C = 1450 \) for the combined cost of the sofa and coffee table.

Step 3: Substitute and Simplify

Substitute \( S = 2L \) into both equations. For the total cost: \( 2L + L + C = 2050 \), which simplifies to \( 3L + C = 2050 \). For the sofa and table: \( 2L + C = 1450 \).

Step 4: Solve for C

Using the equation \( 2L + C = 1450 \), solve for \( C \): \( C = 1450 - 2L \).

Step 5: Substitute C into Total Cost Equation

Substitute \( C = 1450 - 2L \) into the equation \( 3L + C = 2050 \): \( 3L + (1450 - 2L) = 2050 \). Simplify to \( L + 1450 = 2050 \).

Step 6: Solve for L

Subtract 1450 from both sides: \( L = 2050 - 1450 \). Thus, \( L = 600 \).

Step 7: Solve for S

Since \( S = 2L \), substitute \( L = 600 \): \( S = 2 \times 600 = 1200 \).

Step 8: Solve for C

Substitute \( L = 600 \) into \( C = 1450 - 2L \): \( C = 1450 - 1200 = 250 \).

Step 9: Verify the Solution

Check the total cost: \( S + L + C = 1200 + 600 + 250 = 2050 \), which matches the given total. Thus, the values are correct.

Key Concepts

Variables

Variables are like placeholders or symbols that represent unknown values within a problem. In solving systems of equations, variables are essential because they allow us to define and identify different elements we want to figure out.
In this exercise:

• Let the cost of the love seat be represented by the variable \( L \).
• The sofa's cost is denoted by \( S \).
• The cost of the coffee table is written as \( C \).

Using variables, we can express relationships and equations concisely. For instance, knowing that the sofa costs twice as much as the love seat allows us to define \( S = 2L \). This simplifies further calculations.

Equations

Equations are statements that show the equality between two expressions. They are pivotal in finding unknown values in mathematical problems.
Through the exercise, two main equations surface based on the problem's conditions:

• The total cost equation: \( S + L + C = 2050 \). This equation accounts for the entire combined cost of the furniture pieces.
• The equation for the sofa and coffee table combined: \( S + C = 1450 \). This focuses on their combined price only.

These equations provide a framework from which you can start solving for the unknown variables. By representing real-world relationships in a mathematical format, equations form a bridge between the problem description and its solution.

Substitution Method

The substitution method is a powerful tool in solving systems of equations. It involves replacing one variable with an expression in terms of another variable.
In our case, after identifying \( S = 2L \), we can substitute \( S \) in both equations. This results in:

• For the total cost: \( 2L + L + C = 2050 \) simplifies to \( 3L + C = 2050 \).
• For the sofa and coffee table: \( 2L + C = 1450 \).

The substitution method helps reduce the number of variables in an equation, making it easier to isolate and solve for one variable at a time. This approach brings clarity, as you work through each equation, paving the way towards finding the unknown values.

Algebraic Solution

An algebraic solution involves performing calculations to find the specific values of the variables. After setting up your equations and performing substitution, the following steps lead to the solution:

• Solve for \( C \) from \( 2L + C = 1450 \): \( C = 1450 - 2L \).
• Substitute \( C \) back into the total cost equation: \( 3L + (1450 - 2L) = 2050 \).
• Simplify to solve for \( L \): \( L + 1450 = 2050 \), thus \( L = 600 \).

Once \( L \) is known, determine \( S \) and \( C \):

• Calculate \( S = 2L = 1200 \).
• Substitute \( L \) into \( C = 1450 - 2L \) to find \( C = 250 \).

This systematic approach ensures that each piece of furniture's cost matches the problem's constraints, confirming the solution's accuracy.