Answer :
Let's break down the problem step by step to determine the appropriate equation that represents Ali's saving situation.
### Step 1: Understanding the Situation
- Ali currently has \(\$ 25\). This is the initial amount of money he has before he starts saving, which we can refer to as the initial money, \(y_0\).
- Every week, Ali saves \(\$ 5\). This is the amount he saves per week, which we can refer to as the savings per week, \(s\).
### Step 2: Formulating the Equation
We need to develop a mathematical equation that shows Ali's total savings (\(y\)) over time in weeks (\(x\)).
1. Initial Savings: Ali starts with \(\$ 25\). So, if \(x = 0\), \(y = 25\).
2. Weekly Savings: Each week, Ali saves \(\$ 5\). This means for every week \(x\), the total savings increase by \(5x\).
Combining these two pieces of information, the equation that represents Ali’s total savings over \(x\) weeks can be written as:
[tex]\[ y = y_0 + s \cdot x \][/tex]
Given that:
- \(y_0 = 25\)
- \(s = 5\)
Substitute these values into the equation:
[tex]\[ y = 25 + 5x \][/tex]
### Step 3: Verification With Given Options
Now let's match this derived equation with the given options:
- \( y = 25x - 5 \) - This equation is incorrect because it doesn't represent the initial savings and adds a term that incorrectly subtracts savings.
- \( y = 5x + 25 \) - This equation is correct. It accurately represents the situation where \(y\) is the total savings, which starts at 25, and increases by 5 for each week \(x\).
- \( y = 25x + 5 \) - This equation is incorrect because it implies an initial savings amount of 5 and a weekly increase of 25.
- \( y = 5x - 25 \) - This equation is incorrect because it subtracts from the total savings improperly.
Thus, the correct equation is:
[tex]\[ y = 5x + 25 \][/tex]
### Step 4: Determine if It’s a Function
An equation represents a function if each input (value of \(x\)) corresponds to exactly one output (value of \(y\)).
In this case:
- For every value of \(x\) (number of weeks), there is a unique corresponding value of \(y\) (total savings).
Hence, this equation does indeed represent a function.
### Conclusion
- The equation that represents Ali's saving situation is \( y = 5x + 25 \).
- Yes, this equation represents a function.
### Step 1: Understanding the Situation
- Ali currently has \(\$ 25\). This is the initial amount of money he has before he starts saving, which we can refer to as the initial money, \(y_0\).
- Every week, Ali saves \(\$ 5\). This is the amount he saves per week, which we can refer to as the savings per week, \(s\).
### Step 2: Formulating the Equation
We need to develop a mathematical equation that shows Ali's total savings (\(y\)) over time in weeks (\(x\)).
1. Initial Savings: Ali starts with \(\$ 25\). So, if \(x = 0\), \(y = 25\).
2. Weekly Savings: Each week, Ali saves \(\$ 5\). This means for every week \(x\), the total savings increase by \(5x\).
Combining these two pieces of information, the equation that represents Ali’s total savings over \(x\) weeks can be written as:
[tex]\[ y = y_0 + s \cdot x \][/tex]
Given that:
- \(y_0 = 25\)
- \(s = 5\)
Substitute these values into the equation:
[tex]\[ y = 25 + 5x \][/tex]
### Step 3: Verification With Given Options
Now let's match this derived equation with the given options:
- \( y = 25x - 5 \) - This equation is incorrect because it doesn't represent the initial savings and adds a term that incorrectly subtracts savings.
- \( y = 5x + 25 \) - This equation is correct. It accurately represents the situation where \(y\) is the total savings, which starts at 25, and increases by 5 for each week \(x\).
- \( y = 25x + 5 \) - This equation is incorrect because it implies an initial savings amount of 5 and a weekly increase of 25.
- \( y = 5x - 25 \) - This equation is incorrect because it subtracts from the total savings improperly.
Thus, the correct equation is:
[tex]\[ y = 5x + 25 \][/tex]
### Step 4: Determine if It’s a Function
An equation represents a function if each input (value of \(x\)) corresponds to exactly one output (value of \(y\)).
In this case:
- For every value of \(x\) (number of weeks), there is a unique corresponding value of \(y\) (total savings).
Hence, this equation does indeed represent a function.
### Conclusion
- The equation that represents Ali's saving situation is \( y = 5x + 25 \).
- Yes, this equation represents a function.
