To determine the correct equation representing the amount of money \( y \) in Sonia's savings account after \( x \) weeks, we can analyze the information provided.
After 5 weeks, the total in the account is $45: \[ y = 45 \quad \text{when} \quad x = 5 \]
After 10 weeks, the total in the account is $70: \[ y = 70 \quad \text{when} \quad x = 10 \]
Let's denote the amount added each week as \( a \). Therefore, the total amount in the savings account can be represented by the equation: \[ y = a \cdot x + C \] where \( C \) is the initial amount in the account.
Step 1: Set up equations
Using the information from the 5th and 10th weeks, we can create the following equations:
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For \( x = 5 \): \[ y = a \cdot 5 + C = 45 \quad \text{(1)} \]
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For \( x = 10 \): \[ y = a \cdot 10 + C = 70 \quad \text{(2)} \]
Step 2: Subtract equations to eliminate \( C \)
Subtract equation (1) from equation (2): \[ (a \cdot 10 + C) - (a \cdot 5 + C) = 70 - 45 \]
This simplifies to: \[ 5a = 25 \]
Solving for \( a \): \[ a = 5 \]
Step 3: Substitute \( a \) back to find \( C \)
Now that we have \( a \), we can substitute it back into equation (1) to find \( C \): \[ 5 \cdot 5 + C = 45 \] \[ 25 + C = 45 \] \[ C = 20 \]
Step 4: Write the equation for \( y \)
We have found \( a = 5 \) and \( C = 20 \). Now we can write the equation for \( y \): \[ y = 5x + 20 \]
Select the correct option:
Based on the options provided, the correct equation is: B \( y = 5x + 20 \)