Use The Table To Answer The Question.${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 6 & -7 \ \hline 1 & 3 \ \hline -2 & 9 \ \hline \end{tabular} }$What Is The Equation Of The Line That Passes Through The Points Given In The Table?A.
Understanding the Problem
To find the equation of a line that passes through the points given in the table, we need to use the concept of linear equations and the slope-intercept form. The slope-intercept form of a linear equation is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Identifying the Slope and Y-Intercept
To find the equation of the line, we need to calculate the slope (m) and the y-intercept (b) using the points given in the table.
| x | y |
|---|---|
| 6 | -7 |
| 1 | 3 |
| -2 | 9 |
We can use any two points to calculate the slope. Let's use the points (6, -7) and (1, 3).
Calculating the Slope
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points.
m = (3 - (-7)) / (1 - 6) m = (3 + 7) / (-5) m = 10 / (-5) m = -2
Calculating the Y-Intercept
Now that we have the slope (m), we can use one of the points to calculate the y-intercept (b). Let's use the point (6, -7).
y = mx + b -7 = (-2)(6) + b -7 = -12 + b b = 5
Writing the Equation of the Line
Now that we have the slope (m) and the y-intercept (b), we can write the equation of the line in slope-intercept form.
y = mx + b y = (-2)x + 5
Conclusion
In this article, we used a table to find the equation of a line that passes through the points given in the table. We calculated the slope (m) and the y-intercept (b) using the points (6, -7) and (1, 3). The equation of the line is y = (-2)x + 5.
Example Use Cases
- Finding the equation of a line that passes through two points.
- Calculating the slope and y-intercept of a line.
- Writing the equation of a line in slope-intercept form.
Step-by-Step Solution
- Identify the points given in the table.
- Choose two points to calculate the slope.
- Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1).
- Use one of the points to calculate the y-intercept (b).
- Write the equation of the line in slope-intercept form.
Tips and Tricks
- Make sure to choose two points that are not the same.
- Use the correct formula to calculate the slope.
- Check your calculations to ensure that the equation of the line is correct.
Common Mistakes
- Choosing two points that are the same.
- Using the wrong formula to calculate the slope.
- Not checking calculations to ensure that the equation of the line is correct.
Real-World Applications
- Finding the equation of a line that passes through two points in a coordinate plane.
- Calculating the slope and y-intercept of a line in a real-world scenario.
- Writing the equation of a line in slope-intercept form to model a real-world situation.
Conclusion
In this article, we used a table to find the equation of a line that passes through the points given in the table. We calculated the slope (m) and the y-intercept (b) using the points (6, -7) and (1, 3). The equation of the line is y = (-2)x + 5. We also discussed example use cases, step-by-step solutions, tips and tricks, common mistakes, and real-world applications.
Q: What is the equation of a line?
A: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of points on the line. It is typically written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: How do I find the equation of a line that passes through two points?
A: To find the equation of a line that passes through two points, you need to calculate the slope (m) and the y-intercept (b) using the points. You can use the formula m = (y2 - y1) / (x2 - x1) to calculate the slope, and then use one of the points to calculate the y-intercept.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is typically denoted by the letter m.
Q: How do I calculate the slope of a line?
A: To calculate the slope of a line, you need to use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points on the line.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.
Q: How do I calculate the y-intercept of a line?
A: To calculate the y-intercept of a line, you need to use one of the points on the line and the slope (m) to solve for b in the equation y = mx + b.
Q: Can I use any two points to calculate the equation of a line?
A: Yes, you can use any two points to calculate the equation of a line. However, make sure that the two points are not the same, as this would result in a vertical line.
Q: What is the difference between the slope and the y-intercept?
A: The slope (m) is a measure of how steep the line is, while the y-intercept (b) is the point where the line intersects the y-axis.
Q: How do I write the equation of a line in slope-intercept form?
A: To write the equation of a line in slope-intercept form, you need to use the slope (m) and the y-intercept (b) in the equation y = mx + b.
Q: Can I use the equation of a line to solve real-world problems?
A: Yes, you can use the equation of a line to solve real-world problems. For example, you can use the equation of a line to model the relationship between two variables, such as the cost of a product and the quantity sold.
Q: What are some common mistakes to avoid when finding the equation of a line?
A: Some common mistakes to avoid when finding the equation of a line include:
- Choosing two points that are the same
- Using the wrong formula to calculate the slope
- Not checking calculations to ensure that the equation of the line is correct
Q: How do I check my calculations to ensure that the equation of the line is correct?
A: To check your calculations, you can plug in the values of x and y into the equation of the line and verify that the equation is true.
Q: Can I use a calculator to find the equation of a line?
A: Yes, you can use a calculator to find the equation of a line. However, make sure to check your calculations to ensure that the equation is correct.
Q: What are some real-world applications of the equation of a line?
A: Some real-world applications of the equation of a line include:
- Modeling the relationship between two variables, such as the cost of a product and the quantity sold
- Calculating the slope and y-intercept of a line in a real-world scenario
- Writing the equation of a line in slope-intercept form to model a real-world situation
Q: How do I use the equation of a line to solve real-world problems?
A: To use the equation of a line to solve real-world problems, you need to:
- Identify the variables and the relationship between them
- Write the equation of the line in slope-intercept form
- Use the equation to solve for the unknown variable
Q: Can I use the equation of a line to solve problems that involve more than two variables?
A: Yes, you can use the equation of a line to solve problems that involve more than two variables. However, you need to use a more complex equation, such as a quadratic equation or a system of linear equations.
Q: What are some common real-world applications of the equation of a line?
A: Some common real-world applications of the equation of a line include:
- Modeling the relationship between the cost of a product and the quantity sold
- Calculating the slope and y-intercept of a line in a real-world scenario
- Writing the equation of a line in slope-intercept form to model a real-world situation
Q: How do I use the equation of a line to model real-world situations?
A: To use the equation of a line to model real-world situations, you need to:
- Identify the variables and the relationship between them
- Write the equation of the line in slope-intercept form
- Use the equation to model the real-world situation
Q: Can I use the equation of a line to solve problems that involve time and distance?
A: Yes, you can use the equation of a line to solve problems that involve time and distance. For example, you can use the equation of a line to model the relationship between the distance traveled and the time taken.
Q: What are some common mistakes to avoid when using the equation of a line to solve real-world problems?
A: Some common mistakes to avoid when using the equation of a line to solve real-world problems include:
- Not identifying the variables and the relationship between them
- Not writing the equation of the line in slope-intercept form
- Not checking calculations to ensure that the equation is correct
Q: How do I check my calculations to ensure that the equation of the line is correct when using it to solve real-world problems?
A: To check your calculations, you can plug in the values of x and y into the equation of the line and verify that the equation is true.
Q: Can I use the equation of a line to solve problems that involve more than one variable?
A: Yes, you can use the equation of a line to solve problems that involve more than one variable. However, you need to use a more complex equation, such as a quadratic equation or a system of linear equations.
Q: What are some common real-world applications of the equation of a line that involve more than one variable?
A: Some common real-world applications of the equation of a line that involve more than one variable include:
- Modeling the relationship between the cost of a product and the quantity sold
- Calculating the slope and y-intercept of a line in a real-world scenario
- Writing the equation of a line in slope-intercept form to model a real-world situation
Q: How do I use the equation of a line to solve problems that involve more than one variable?
A: To use the equation of a line to solve problems that involve more than one variable, you need to:
- Identify the variables and the relationship between them
- Write the equation of the line in slope-intercept form
- Use the equation to solve for the unknown variable
Q: Can I use the equation of a line to solve problems that involve non-linear relationships?
A: Yes, you can use the equation of a line to solve problems that involve non-linear relationships. However, you need to use a more complex equation, such as a quadratic equation or a system of linear equations.
Q: What are some common real-world applications of the equation of a line that involve non-linear relationships?
A: Some common real-world applications of the equation of a line that involve non-linear relationships include:
- Modeling the relationship between the cost of a product and the quantity sold
- Calculating the slope and y-intercept of a line in a real-world scenario
- Writing the equation of a line in slope-intercept form to model a real-world situation
Q: How do I use the equation of a line to solve problems that involve non-linear relationships?
A: To use the equation of a line to solve problems that involve non-linear relationships, you need to:
- Identify the variables and the relationship between them
- Write the equation of the line in slope-intercept form
- Use the equation to solve for the unknown variable
Q: Can I use the equation of a line to solve problems that involve multiple variables and non-linear relationships?
A: Yes, you can use the equation of a line to solve problems that involve multiple variables and non-linear relationships. However, you need to use a more complex equation, such as a quadratic equation or a system of linear equations.