Loren Solved The Equation $10=\frac{19}{9}(149)+b$ For $b$ As Part Of Her Work To Find The Equation Of A Trend Line That Passes Through The Points $(1,130)$ And $ ( 10 , 149 ) (10,149) ( 10 , 149 ) [/tex]. What Error Did Loren
Introduction
In mathematics, solving equations is a crucial step in understanding various concepts, including trend lines. Loren, in her work, aimed to find the equation of a trend line that passes through the points (1,130) and (10,149). To achieve this, she solved the equation 10 = (19/9)(149) + b for b. However, in this analysis, we will identify the error Loren made in her calculation.
The Equation
The given equation is 10 = (19/9)(149) + b. To solve for b, we need to follow the order of operations (PEMDAS):
- Multiply 19/9 by 149
- Add b to the result
Step 1: Multiply 19/9 by 149
To multiply 19/9 by 149, we can use the distributive property:
(19/9)(149) = (19 × 149) / 9
Now, let's calculate the numerator:
19 × 149 = 2831
So, the equation becomes:
10 = 2831 / 9 + b
Step 2: Divide 2831 by 9
To divide 2831 by 9, we can use long division or a calculator:
2831 ÷ 9 = 314.5556 (approximately)
Now, the equation becomes:
10 = 314.5556 + b
Step 3: Solve for b
To solve for b, we need to isolate b on one side of the equation. We can do this by subtracting 314.5556 from both sides:
b = 10 - 314.5556
b ≈ -304.5556
The Error
Loren's error lies in the calculation of (19/9)(149). She did not follow the order of operations correctly. To find the correct value of b, we need to multiply 19/9 by 149 and then add b to the result.
Correct Calculation
Let's redo the calculation:
(19/9)(149) = (19 × 149) / 9
19 × 149 = 2831
2831 ÷ 9 = 314.5556 (approximately)
Now, the equation becomes:
10 = 314.5556 + b
b = 10 - 314.5556
b ≈ -304.5556
Conclusion
In conclusion, Loren's error was in the calculation of (19/9)(149). She did not follow the order of operations correctly, which led to an incorrect value of b. To find the correct value of b, we need to multiply 19/9 by 149 and then add b to the result.
Recommendations
To avoid similar errors in the future, it's essential to follow the order of operations (PEMDAS) and double-check calculations. Additionally, using a calculator or a computer algebra system can help reduce errors and increase accuracy.
Future Work
In future work, Loren can use the correct value of b to find the equation of the trend line that passes through the points (1,130) and (10,149). She can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is a point on the line.
Slope and Equation
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (1, 130) and (x2, y2) = (10, 149)
m = (149 - 130) / (10 - 1)
m = 19 / 9
Now, we can use the point-slope form to find the equation of the trend line:
y - 130 = (19/9)(x - 1)
y - 130 = (19/9)x - 19/9
y = (19/9)x + 130 - 19/9
y = (19/9)x + 1191/9
Final Equation
The final equation of the trend line is:
y = (19/9)x + 1191/9
Introduction
In our previous article, we analyzed Loren's equation and identified the error she made in her calculation. We also provided the correct calculation and the final equation of the trend line that passes through the points (1,130) and (10,149). In this article, we will answer some frequently asked questions related to Loren's equation and provide additional insights.
Q&A
Q: What is the error Loren made in her calculation?
A: Loren's error was in the calculation of (19/9)(149). She did not follow the order of operations correctly, which led to an incorrect value of b.
Q: How can we avoid similar errors in the future?
A: To avoid similar errors in the future, it's essential to follow the order of operations (PEMDAS) and double-check calculations. Additionally, using a calculator or a computer algebra system can help reduce errors and increase accuracy.
Q: What is the correct value of b?
A: The correct value of b is approximately -304.5556.
Q: How can we find the equation of the trend line that passes through the points (1,130) and (10,149)?
A: To find the equation of the trend line, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is a point on the line.
Q: What is the slope of the trend line?
A: The slope of the trend line is 19/9.
Q: What is the final equation of the trend line?
A: The final equation of the trend line is:
y = (19/9)x + 1191/9
Q: What is the significance of the trend line?
A: The trend line represents the relationship between the independent variable (x) and the dependent variable (y). In this case, the trend line passes through the points (1,130) and (10,149), indicating a linear relationship between the variables.
Q: How can we use the trend line in real-world applications?
A: The trend line can be used in various real-world applications, such as predicting future values, identifying patterns, and making informed decisions.
Additional Insights
- The trend line can be used to model real-world phenomena, such as population growth, economic trends, and weather patterns.
- The slope of the trend line represents the rate of change between the independent variable and the dependent variable.
- The y-intercept of the trend line represents the value of the dependent variable when the independent variable is zero.
Conclusion
In conclusion, Loren's equation and the trend line that passes through the points (1,130) and (10,149) provide valuable insights into the relationship between the independent variable and the dependent variable. By following the order of operations and using the correct calculations, we can avoid errors and increase accuracy. The trend line can be used in various real-world applications, and its significance lies in its ability to model real-world phenomena and make informed decisions.
Recommendations
- Use the trend line to model real-world phenomena and make informed decisions.
- Double-check calculations and follow the order of operations to avoid errors.
- Use a calculator or a computer algebra system to reduce errors and increase accuracy.
- Explore the significance of the trend line and its applications in real-world scenarios.