Assuming A Linear Relationship, Find The Missing Value In The Table Below.${ \begin{tabular}{|c|c|c|c|c|c|} \hline X X X & 1 & 2 & 3 & 4 & 5 \ \hline Y Y Y & 16 & 25 & 34 & 43 & \ \hline \end{tabular} }$Answer:
Introduction
In mathematics, a linear relationship is a relationship between two variables where one variable is a constant multiple of the other. In other words, if we have two variables x and y, a linear relationship between them means that y can be expressed as a linear function of x, i.e., y = mx + b, where m is the slope and b is the y-intercept. In this article, we will assume a linear relationship between the variables x and y in a given table and find the missing value.
Understanding the Table
The table below shows the values of x and y for five different values of x.
| x | y |
|---|---|
| 1 | 16 |
| 2 | 25 |
| 3 | 34 |
| 4 | 43 |
| 5 | ? |
Finding the Missing Value
To find the missing value of y for x = 5, we need to first find the slope (m) of the linear relationship between x and y. We can do this by finding the difference in y-values for a given difference in x-values.
Let's find the difference in y-values for a difference of 1 in x-values.
| x | y | Δx | Δy |
|---|---|---|---|
| 1 | 16 | 1 | 9 |
| 2 | 25 | 1 | 10 |
| 3 | 34 | 1 | 9 |
| 4 | 43 | 1 | 9 |
As we can see, the difference in y-values is constant for a difference of 1 in x-values, which means that the slope (m) is constant.
Now, let's find the slope (m) by dividing the difference in y-values by the difference in x-values.
m = Δy / Δx = 9 / 1 = 9
Finding the Equation of the Line
Now that we have the slope (m), we can find the equation of the line using the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line.
Let's use the point (1, 16) to find the equation of the line.
y - 16 = 9(x - 1)
Simplifying the equation, we get:
y = 9x + 5
Finding the Missing Value
Now that we have the equation of the line, we can find the missing value of y for x = 5 by plugging in x = 5 into the equation.
y = 9(5) + 5 y = 45 + 5 y = 50
Therefore, the missing value of y for x = 5 is 50.
Conclusion
In this article, we assumed a linear relationship between the variables x and y in a given table and found the missing value of y for x = 5. We first found the slope (m) of the linear relationship by finding the difference in y-values for a given difference in x-values. Then, we found the equation of the line using the point-slope form of a linear equation. Finally, we plugged in x = 5 into the equation to find the missing value of y.
Example Problems
- Find the missing value of y for x = 6 in the table below.
| x | y |
|---|---|
| 1 | 10 |
| 2 | 15 |
| 3 | 20 |
| 4 | 25 |
| 5 | 30 |
| 6 | ? |
- Find the missing value of y for x = 3 in the table below.
| x | y |
|---|---|
| 1 | 5 |
| 2 | 10 |
| 3 | ? |
| 4 | 20 |
| 5 | 25 |
Solutions
- To find the missing value of y for x = 6, we need to first find the slope (m) of the linear relationship between x and y. We can do this by finding the difference in y-values for a given difference in x-values.
| x | y | Δx | Δy |
|---|---|---|---|
| 1 | 10 | 1 | 5 |
| 2 | 15 | 1 | 5 |
| 3 | 20 | 1 | 5 |
| 4 | 25 | 1 | 5 |
| 5 | 30 | 1 | 5 |
As we can see, the difference in y-values is constant for a difference of 1 in x-values, which means that the slope (m) is constant.
Now, let's find the slope (m) by dividing the difference in y-values by the difference in x-values.
m = Δy / Δx = 5 / 1 = 5
Now that we have the slope (m), we can find the equation of the line using the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line.
Let's use the point (1, 10) to find the equation of the line.
y - 10 = 5(x - 1)
Simplifying the equation, we get:
y = 5x + 5
Now that we have the equation of the line, we can find the missing value of y for x = 6 by plugging in x = 6 into the equation.
y = 5(6) + 5 y = 30 + 5 y = 35
Therefore, the missing value of y for x = 6 is 35.
- To find the missing value of y for x = 3, we need to first find the slope (m) of the linear relationship between x and y. We can do this by finding the difference in y-values for a given difference in x-values.
| x | y | Δx | Δy |
|---|---|---|---|
| 1 | 5 | 1 | 5 |
| 2 | 10 | 1 | 5 |
| 3 | ? | 1 | ? |
| 4 | 20 | 1 | 10 |
| 5 | 25 | 1 | 5 |
As we can see, the difference in y-values is not constant for a difference of 1 in x-values, which means that the slope (m) is not constant.
However, we can see that the difference in y-values is constant for a difference of 2 in x-values.
| x | y | Δx | Δy |
|---|---|---|---|
| 1 | 5 | 2 | 10 |
| 3 | ? | 2 | ? |
| 4 | 20 | 2 | 10 |
| 5 | 25 | 2 | 5 |
Now, let's find the slope (m) by dividing the difference in y-values by the difference in x-values.
m = Δy / Δx = 10 / 2 = 5
Now that we have the slope (m), we can find the equation of the line using the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line.
Let's use the point (1, 5) to find the equation of the line.
y - 5 = 5(x - 1)
Simplifying the equation, we get:
y = 5x + 0
Now that we have the equation of the line, we can find the missing value of y for x = 3 by plugging in x = 3 into the equation.
y = 5(3) + 0 y = 15 + 0 y = 15
Q: What is a linear relationship?
A: A linear relationship is a relationship between two variables where one variable is a constant multiple of the other. In other words, if we have two variables x and y, a linear relationship between them means that y can be expressed as a linear function of x, i.e., y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the missing value in a table with a linear relationship?
A: To find the missing value in a table with a linear relationship, you need to first find the slope (m) of the linear relationship by finding the difference in y-values for a given difference in x-values. Then, you can find the equation of the line using the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line. Finally, you can plug in the x-value of the missing value into the equation to find the missing value.
Q: What if the difference in y-values is not constant for a given difference in x-values?
A: If the difference in y-values is not constant for a given difference in x-values, it means that the slope (m) is not constant. In this case, you need to find the difference in y-values for a different difference in x-values. For example, if the difference in y-values is constant for a difference of 2 in x-values, you can use this to find the slope (m) and the equation of the line.
Q: How do I know if the relationship is linear or not?
A: To determine if the relationship is linear or not, you need to check if the difference in y-values is constant for a given difference in x-values. If it is, then the relationship is linear. If it's not, then the relationship is not linear.
Q: Can I use this method to find the missing value in a table with a non-linear relationship?
A: No, this method only works for tables with a linear relationship. If the relationship is non-linear, you need to use a different method to find the missing value.
Q: What if I have a table with multiple missing values?
A: If you have a table with multiple missing values, you can use the same method to find each missing value individually. However, you need to make sure that the relationship is linear for each missing value.
Q: Can I use this method to find the missing value in a table with a large number of rows?
A: Yes, this method can be used to find the missing value in a table with a large number of rows. However, you need to make sure that the relationship is linear for each missing value.
Q: Is there a shortcut to find the missing value in a table with a linear relationship?
A: Yes, there is a shortcut to find the missing value in a table with a linear relationship. You can use the formula: y = mx + b, where m is the slope and b is the y-intercept. If you know the slope (m) and the y-intercept (b), you can plug in the x-value of the missing value into the formula to find the missing value.
Q: How do I find the slope (m) and the y-intercept (b) of a linear relationship?
A: To find the slope (m) and the y-intercept (b) of a linear relationship, you need to use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line. You can use any point on the line to find the slope (m) and the y-intercept (b).
Q: Can I use this method to find the missing value in a table with a linear relationship that has a negative slope?
A: Yes, this method can be used to find the missing value in a table with a linear relationship that has a negative slope. However, you need to make sure that the relationship is linear for each missing value.
Q: Is there a limit to the number of missing values that can be found using this method?
A: No, there is no limit to the number of missing values that can be found using this method. However, you need to make sure that the relationship is linear for each missing value.
Q: Can I use this method to find the missing value in a table with a linear relationship that has a fractional slope?
A: Yes, this method can be used to find the missing value in a table with a linear relationship that has a fractional slope. However, you need to make sure that the relationship is linear for each missing value.
Q: Is there a way to check if the relationship is linear before finding the missing value?
A: Yes, there is a way to check if the relationship is linear before finding the missing value. You can use the difference in y-values for a given difference in x-values to check if the relationship is linear. If the difference in y-values is constant for a given difference in x-values, then the relationship is linear.