Answer :
To determine which of the potential points will lie on the line that represents Steven's savings account balance over time, given the monthly balances, we need to find the equation of the line representing Steven's savings account balance.
Here are the steps to solve this problem:
1. Identify the given data points:
[tex]\[ (1, 400), (3, 1200), (7, 2800), (9, 3600), (13, 5200) \][/tex]
2. Calculate the slope (rate of change) between each consecutive data point:
- Between [tex]\( (1, 400) \)[/tex] and [tex]\( (3, 1200) \)[/tex]:
[tex]\[ \text{slope}_1 = \frac{1200 - 400}{3 - 1} = \frac{800}{2} = 400 \][/tex]
- Between [tex]\( (3, 1200) \)[/tex] and [tex]\( (7, 2800) \)[/tex]:
[tex]\[ \text{slope}_2 = \frac{2800 - 1200}{7 - 3} = \frac{1600}{4} = 400 \][/tex]
- Between [tex]\( (7, 2800) \)[/tex] and [tex]\( (9, 3600) \)[/tex]:
[tex]\[ \text{slope}_3 = \frac{3600 - 2800}{9 - 7} = \frac{800}{2} = 400 \][/tex]
- Between [tex]\( (9, 3600) \)[/tex] and [tex]\( (13, 5200) \)[/tex]:
[tex]\[ \text{slope}_4 = \frac{5200 - 3600}{13 - 9} = \frac{1600}{4} = 400 \][/tex]
Since all the slopes are consistent, the rate of change is [tex]\( 400 \)[/tex].
3. Determine the average slope (if needed):
[tex]\[ \text{avg\_slope} = \frac{400 + 400 + 400 + 400}{4} = 400 \][/tex]
4. Use one of the points [tex]\((1, 400)\)[/tex] to find the y-intercept of the line:
- Slope-intercept form of the line:
[tex]\[ y = mx + c \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
- Substitute one point and the average slope into the equation:
[tex]\[ 400 = 400 \cdot 1 + c \][/tex]
[tex]\[ 400 = 400 + c \Rightarrow c = 0 \][/tex]
- Therefore, the equation of the line is:
[tex]\[ y = 400x + 0 \Rightarrow y = 400x \][/tex]
5. Check which given potential points lie on the line [tex]\( y = 400x \)[/tex]:
- Point [tex]\( (22, 6400) \)[/tex]:
[tex]\[ y = 400 \cdot 22 = 8800 \quad \text{(not equal to 6400)} \][/tex]
- Point [tex]\( (23, 9200) \)[/tex]:
[tex]\[ y = 400 \cdot 23 = 9200 \quad \text{(equal to 9200, lies on the line)} \][/tex]
- Point [tex]\( (24, 8800) \)[/tex]:
[tex]\[ y = 400 \cdot 24 = 9600 \quad \text{(not equal to 8800)} \][/tex]
- Point [tex]\( (26, 10400) \)[/tex]:
[tex]\[ y = 400 \cdot 26 = 10400 \quad \text{(equal to 10400, lies on the line)} \][/tex]
- Point [tex]\( (27, 10600) \)[/tex]:
[tex]\[ y = 400 \cdot 27 = 10800 \quad \text{(not equal to 10600)} \][/tex]
6. Conclusion:
The points that will lie on the line representing Steven's balance over time are:
[tex]\[ (23, 9200) \quad \text{and} \quad (26, 10400) \][/tex]
Here are the steps to solve this problem:
1. Identify the given data points:
[tex]\[ (1, 400), (3, 1200), (7, 2800), (9, 3600), (13, 5200) \][/tex]
2. Calculate the slope (rate of change) between each consecutive data point:
- Between [tex]\( (1, 400) \)[/tex] and [tex]\( (3, 1200) \)[/tex]:
[tex]\[ \text{slope}_1 = \frac{1200 - 400}{3 - 1} = \frac{800}{2} = 400 \][/tex]
- Between [tex]\( (3, 1200) \)[/tex] and [tex]\( (7, 2800) \)[/tex]:
[tex]\[ \text{slope}_2 = \frac{2800 - 1200}{7 - 3} = \frac{1600}{4} = 400 \][/tex]
- Between [tex]\( (7, 2800) \)[/tex] and [tex]\( (9, 3600) \)[/tex]:
[tex]\[ \text{slope}_3 = \frac{3600 - 2800}{9 - 7} = \frac{800}{2} = 400 \][/tex]
- Between [tex]\( (9, 3600) \)[/tex] and [tex]\( (13, 5200) \)[/tex]:
[tex]\[ \text{slope}_4 = \frac{5200 - 3600}{13 - 9} = \frac{1600}{4} = 400 \][/tex]
Since all the slopes are consistent, the rate of change is [tex]\( 400 \)[/tex].
3. Determine the average slope (if needed):
[tex]\[ \text{avg\_slope} = \frac{400 + 400 + 400 + 400}{4} = 400 \][/tex]
4. Use one of the points [tex]\((1, 400)\)[/tex] to find the y-intercept of the line:
- Slope-intercept form of the line:
[tex]\[ y = mx + c \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
- Substitute one point and the average slope into the equation:
[tex]\[ 400 = 400 \cdot 1 + c \][/tex]
[tex]\[ 400 = 400 + c \Rightarrow c = 0 \][/tex]
- Therefore, the equation of the line is:
[tex]\[ y = 400x + 0 \Rightarrow y = 400x \][/tex]
5. Check which given potential points lie on the line [tex]\( y = 400x \)[/tex]:
- Point [tex]\( (22, 6400) \)[/tex]:
[tex]\[ y = 400 \cdot 22 = 8800 \quad \text{(not equal to 6400)} \][/tex]
- Point [tex]\( (23, 9200) \)[/tex]:
[tex]\[ y = 400 \cdot 23 = 9200 \quad \text{(equal to 9200, lies on the line)} \][/tex]
- Point [tex]\( (24, 8800) \)[/tex]:
[tex]\[ y = 400 \cdot 24 = 9600 \quad \text{(not equal to 8800)} \][/tex]
- Point [tex]\( (26, 10400) \)[/tex]:
[tex]\[ y = 400 \cdot 26 = 10400 \quad \text{(equal to 10400, lies on the line)} \][/tex]
- Point [tex]\( (27, 10600) \)[/tex]:
[tex]\[ y = 400 \cdot 27 = 10800 \quad \text{(not equal to 10600)} \][/tex]
6. Conclusion:
The points that will lie on the line representing Steven's balance over time are:
[tex]\[ (23, 9200) \quad \text{and} \quad (26, 10400) \][/tex]
