Let's analyze the problem step-by-step to determine the correct system of linear equations for the cost of each table (x) and each chair (y).
From the provided information, we can formulate two equations based on the quantities of tables and chairs:
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From the first statement: "5 small tables with 8 chairs for $115"
- This can be represented by the equation: \[ 5x + 8y = 115 \]
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From the second statement: "3 tables with 5 chairs cost $70"
- This can be represented by the equation: \[ 3x + 5y = 70 \]
Thus, the correct system of linear equations is: \[ \begin{cases} 5x + 8y = 115 \ 3x + 5y = 70 \end{cases} \]
Now let's match this with the options provided:
- Option B: \(5x + 8y = 115; 3x + 5y = 70\) is the correct choice.
Question 1: The correct system of linear equations
Answer: B \(5x + 8y = 115; 3x + 5y = 70\)
Question 2: Solving the System of Equations
To find the values of \( x \) (cost of each table) and \( y \) (cost of each chair), we can use the two equations we established above:
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From the first equation: \[ 5x + 8y = 115 \tag{1} \]
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From the second equation: \[ 3x + 5y = 70 \tag{2} \]
Let's solve for one variable. We'll solve equation (1) for \( y \): \[ 8y = 115 - 5x \quad \Rightarrow \quad y = \frac{115 - 5x}{8} \tag{3} \]
Now, substitute equation (3) into equation (2): \[ 3x + 5\left(\frac{115 - 5x}{8}\right) = 70 \] Multiply everything by 8 to eliminate the fraction: \[ 24x + 5(115 - 5x) = 560 \] Expand and simplify: \[ 24x + 575 - 25x = 560 \] Combine like terms: \[ -1x + 575 = 560 \] Rearrange to find \( x \): \[ -1x = 560 - 575 \] \[ -1x = -15 \quad \Rightarrow \quad x = 15 \]
Now substitute \( x = 15 \) back into equation (3) to find \( y \): \[ y = \frac{115 - 5(15)}{8} \] \[ y = \frac{115 - 75}{8} \] \[ y = \frac{40}{8} = 5 \]
Thus, the cost of each table (\( x \)) is $15, and the cost of each chair (\( y \)) is $5.
Question 2: The costs
Answer: A \(x = 15; y = 5\)
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