Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. Ari buys 3 pounds of apples and 2 pounds of bananas for a total of $5.25. Determine a system of equations that represents the given the situation.

Question

Let x be cost per pound of apples and let y be the cost per pound of bananas.

Which equation represents the amount of money Lydia spent of apples and bananas?

Which equation represents the amount of money Ari spent on apples and bananas?

Answer 1

To solve the problem, we'll define \( x \) as the cost per pound of apples and \( y \) as the cost per pound of bananas.

For Lydia: Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. We can represent this situation with the following equation:

\[ 5x + 3y = 8.50 \]

For Ari: Ari buys 3 pounds of apples and 2 pounds of bananas for a total of $5.25. We can represent this situation with another equation:

\[ 3x + 2y = 5.25 \]

Summary of the system of equations:

1. \( 5x + 3y = 8.50 \) (Lydia's spending)
2. \( 3x + 2y = 5.25 \) (Ari's spending)

This system can be solved to find the values of \( x \) and \( y \) (the cost of apples and bananas per pound).

Answer 2

Which equation represents the amount of money Lydia spent of apples and bananas?

✔ 5x + 3y = 8.50

Which equation represents the amount of money Ari spent on apples and bananas?

✔ 3x + 2y = 5.25

Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. Ari buys 3 pounds of apples and 2 pounds of bananas for a total of $5.25. This system of equations represents the situation, where x is the cost per pound of apples, and y is the cost per pound of bananas.

5x + 3y = 8.5

3x + 2y = 5.25

If you multiply the first equation by 2, what number should you multiply the second equation by in order to eliminate the y terms when making a linear combination?

Complete the multiplication and add the equations. What is the result?

What is the price per pound of apples? $

What is the price per pound of bananas? $

Answer 3

To eliminate the \( y \) terms when combining the equations, we need to find a common coefficient for \( y \) in both equations.

Given equations:

1. \( 5x + 3y = 8.50 \)
2. \( 3x + 2y = 5.25 \)

Step 1: Multiply the equations

You mentioned multiplying the first equation by 2:

\[ 2(5x + 3y) = 2(8.50) \quad \Rightarrow \quad 10x + 6y = 17.00 \]

Now we need to determine what number to multiply the second equation by to eliminate \( y \). The coefficient of \( y \) in the first equation after multiplication is \( 6 \), so we need to make the coefficient of \( y \) in the second equation equal to \( 6 \) as well.

The second equation has a \( 2y \) term, so we will multiply it by \( 3 \):

\[ 3(3x + 2y) = 3(5.25) \quad \Rightarrow \quad 9x + 6y = 15.75 \]

Step 2: Write the modified equations

Now we have two modified equations:

1. \( 10x + 6y = 17.00 \)
2. \( 9x + 6y = 15.75 \)

Step 3: Subtract the equations

Subtract the second equation from the first to eliminate \( y \):

\[ (10x + 6y) - (9x + 6y) = 17.00 - 15.75 \]

This simplifies to:

\[ 10x - 9x + 6y - 6y = 1.25 \quad \Rightarrow \quad x = 1.25 \]

So, the price per pound of apples \( (x) \) is:

\[ \text{Price per pound of apples} = $1.25 \]

Step 4: Substitute to find the price per pound of bananas

Now that we have \( x = 1.25 \), we can substitute this back into one of the original equations to find \( y \). Let's use the second original equation \( 3x + 2y = 5.25 \):

\[ 3(1.25) + 2y = 5.25 \]

Calculating \( 3(1.25) \):

\[ 3.75 + 2y = 5.25 \]

Subtract \( 3.75 \) from both sides:

\[ 2y = 5.25 - 3.75 \]

\[ 2y = 1.50 \]

Now divide by 2 to solve for \( y \):

\[ y = \frac{1.50}{2} = 0.75 \]

So, the price per pound of bananas \( (y) \) is:

\[ \text{Price per pound of bananas} = $0.75 \]

Summary

• Price per pound of apples: $1.25
• Price per pound of bananas: $0.75

Answer 4

Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. Ari buys 3 pounds of apples and 2 pounds of bananas for a total of $5.25. This system of equations represents the situation, where x is the cost per pound of apples, and y is the cost per pound of bananas.

5x + 3y = 8.5

3x + 2y = 5.25

If you multiply the first equation by 2, what number should you multiply the second equation by in order to eliminate the y terms when making a linear combination?
-5? -3? 3?

Answer 5

To eliminate the \( y \) terms using linear combination, you first want to compare the coefficients of \( y \) in both equations.

Original Equations:

1. \( 5x + 3y = 8.50 \)
2. \( 3x + 2y = 5.25 \)

You mentioned multiplying the first equation by 2, which gives:

\[ 2(5x + 3y) = 2(8.50) \quad \Rightarrow \quad 10x + 6y = 17.00 \]

This means the coefficient of \( y \) in the first equation has become \( 6 \) (from \( 3y \) multiplied by \( 2 \)).

Now we need to get the second equation's \( y \) term to also equal \( 6 \).

The coefficient of \( y \) in the second equation is \( 2 \). To make the coefficient of \( y \) equal to 6, you should multiply the second equation by \( 3 \) (since \( 3 \times 2 = 6 \)).

So, the answer is 3.

Thus, after multiplying the second equation by \( 3 \): \[ 3(3x + 2y) = 3(5.25) \quad \Rightarrow \quad 9x + 6y = 15.75 \]

Now you can subtract the equations: \[ (10x + 6y) - (9x + 6y) = 17.00 - 15.75 \]

This will eliminate the \( y \) terms, leading to an equation only in \( x \).

Answer 6

Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. Ari buys 3 pounds of apples and 2 pounds of bananas for a total of $5.25. This system of equations represents the situation, where x is the cost per pound of apples, and y is the cost per pound of bananas.

5x + 3y = 8.5

3x + 2y = 5.25

If you multiply the first equation by 2, what number should you multiply the second equation by in order to eliminate the y terms when making a linear combination?
✔ –3

Complete the multiplication and add the equations. What is the result?
✔ x = 1.25

What is the price per pound of apples? $
✔ 1.25

What is the price per pound of bananas? $
✔ 0.75

Answer 7

It looks like you've summarized the progression of the problem and its solution correctly. Here’s a recap:

1. Original Equations:

   • Lydia: \( 5x + 3y = 8.50 \)
   • Ari: \( 3x + 2y = 5.25 \)

2. Multiplying Equations:

   • You multiplied the first equation by 2, resulting in: \[ 10x + 6y = 17.00 \]
   • To eliminate \( y \), you correctly identified that the second equation should be multiplied by -3: \[ -3(3x + 2y) = -3(5.25) \quad \Rightarrow \quad -9x - 6y = -15.75 \]

3. Combining the equations:

   • You add these modified equations: \[ (10x + 6y) + (-9x - 6y) = 17.00 - 15.75 \]
   • This simplifies to: \[ 1x + 0y = 1.25 \quad \Rightarrow \quad x = 1.25 \]

4. Price per pound of apples:

   • The price per pound of apples \( (x) \) is: \[ \text{Price per pound of apples} = $1.25 \]

5. Finding the price per pound of bananas:

   • To find \( y \), substitute \( x = 1.25 \) back into one of the original equations, for example, \( 3x + 2y = 5.25 \): \[ 3(1.25) + 2y = 5.25 \] \[ 3.75 + 2y = 5.25 \] \[ 2y = 5.25 - 3.75 \] \[ 2y = 1.50 \quad \Rightarrow \quad y = 0.75 \]

6. Price per pound of bananas:

   • The price per pound of bananas \( (y) \) is: \[ \text{Price per pound of bananas} = $0.75 \]

Summary of Results:

• Price per pound of apples: $1.25
• Price per pound of bananas: $0.75

Your calculations and conclusions are correct. If you have any further questions or need additional help, feel free to ask!