To solve the problem, we first need to express the given information in the form of linear equations.
From the problem, we have:
-
First equation: 5 small tables with 8 chairs cost $115. This can be expressed as: \[ 5x + 8y = 115 \] where \(x\) is the cost of each table and \(y\) is the cost of each chair.
-
Second equation: 3 tables with 5 chairs cost $70. This can be expressed as: \[ 3x + 5y = 70 \]
Now we have the system of equations:
- \(5x + 8y = 115\)
- \(3x + 5y = 70\)
Now, looking at the options for Question 1, we can conclude that the correct option is: B: \(5x + 8y = 115; , 3x + 5y = 70\)
Next, let's solve the equations to determine the cost of each table (\(x\)) and each chair (\(y\)).
-
From the first equation: \[ 5x + 8y = 115 \quad \text{(1)} \]
-
From the second equation: \[ 3x + 5y = 70 \quad \text{(2)} \]
We can solve this system using substitution or elimination. Here we'll use the elimination method.
Start by multiplying equation (2) by 5 and equation (1) by 3, so we can eliminate \(y\):
-
Multiply (1) by 3: \[ 15x + 24y = 345 \quad \text{(3)} \]
-
Multiply (2) by 5: \[ 15x + 25y = 350 \quad \text{(4)} \]
Now we can set these up as follows: \[ 15x + 24y = 345 \quad \text{(3)} \] \[ 15x + 25y = 350 \quad \text{(4)} \]
Next, subtract equation (3) from equation (4): \[ (15x + 25y) - (15x + 24y) = 350 - 345 \] This simplifies to: \[ y = 5 \]
Now substitute \(y = 5\) back into one of the original equations. We will use equation (1): \[ 5x + 8(5) = 115 \] \[ 5x + 40 = 115 \] \[ 5x = 115 - 40 \] \[ 5x = 75 \] \[ x = 15 \]
Thus, the cost of each table \(x\) is $15 and the cost of each chair \(y\) is $5.
Now, looking at the options for Question 2, the correct answer is: A: \(x = 15; , y = 5\)
In summary:
- The answer to Question 1 is B.
- The answer to Question 2 is A.