Write The Linear Equation That Gives The Rule For This Table.${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 4 & 5 \ \hline 5 & 23 \ \hline 6 & 41 \ \hline 7 & 59 \ \hline \end{tabular} }$Write Your Answer As An Equation With
Introduction
In mathematics, a linear equation is a fundamental concept that represents a relationship between two variables, typically denoted as x and y. It is a crucial tool in algebra and is used to model various real-world scenarios. In this article, we will explore how to write a linear equation that gives the rule for a given table.
Understanding the Table
The table provided consists of five rows, each representing a pair of values for x and y. The values are:
| x | y |
|---|---|
| 4 | 5 |
| 5 | 23 |
| 6 | 41 |
| 7 | 59 |
Identifying the Pattern
To write a linear equation that represents the rule for this table, we need to identify the pattern or relationship between the values of x and y. Upon closer inspection, we can observe that the values of y are increasing by a certain amount for each increment in the value of x.
Calculating the Difference
Let's calculate the difference between the values of y for each increment in x:
| x | y | Difference |
|---|---|---|
| 4 | 5 | - |
| 5 | 23 | 18 |
| 6 | 41 | 18 |
| 7 | 59 | 18 |
We can see that the difference between the values of y is constant at 18 for each increment in x.
Writing the Linear Equation
Now that we have identified the pattern and calculated the difference, we can write a linear equation that represents the rule for this table. The general form of a linear equation is:
y = mx + b
where m is the slope (or rate of change) and b is the y-intercept.
In this case, the slope (m) is 18, since the difference between the values of y is constant at 18 for each increment in x. The y-intercept (b) is 5, since the value of y is 5 when x is 4.
Therefore, the linear equation that represents the rule for this table is:
y = 18x + 5
Verifying the Equation
To verify that this equation represents the rule for the table, we can plug in the values of x and y from the table and check if the equation holds true.
| x | y | y = 18x + 5 |
|---|---|---|
| 4 | 5 | 18(4) + 5 = 72 + 5 = 77 ( incorrect ) |
| 5 | 23 | 18(5) + 5 = 90 + 5 = 95 ( incorrect ) |
| 6 | 41 | 18(6) + 5 = 108 + 5 = 113 ( incorrect ) |
| 7 | 59 | 18(7) + 5 = 126 + 5 = 131 ( incorrect ) |
It appears that the equation y = 18x + 5 does not accurately represent the rule for the table.
Revisiting the Pattern
Let's revisit the pattern and see if we can identify a different relationship between the values of x and y.
| x | y |
|---|---|
| 4 | 5 |
| 5 | 23 |
| 6 | 41 |
| 7 | 59 |
Upon closer inspection, we can observe that the values of y are increasing by a certain amount for each increment in the value of x. Let's calculate the difference between the values of y for each increment in x:
| x | y | Difference |
|---|---|---|
| 4 | 5 | - |
| 5 | 23 | 18 |
| 6 | 41 | 18 |
| 7 | 59 | 18 |
We can see that the difference between the values of y is constant at 18 for each increment in x.
Writing a New Linear Equation
Now that we have identified the pattern and calculated the difference, we can write a new linear equation that represents the rule for this table. The general form of a linear equation is:
y = mx + b
where m is the slope (or rate of change) and b is the y-intercept.
In this case, the slope (m) is 18, since the difference between the values of y is constant at 18 for each increment in x. The y-intercept (b) is 5, since the value of y is 5 when x is 4.
However, we can also observe that the values of y are increasing by 18 for each increment in x, which suggests that the equation is quadratic in nature.
Let's try to write a quadratic equation that represents the rule for this table. The general form of a quadratic equation is:
y = ax^2 + bx + c
where a, b, and c are constants.
We can start by plugging in the values of x and y from the table and solving for a, b, and c.
| x | y |
|---|---|
| 4 | 5 |
| 5 | 23 |
| 6 | 41 |
| 7 | 59 |
Let's start by plugging in the values of x and y from the first row:
5 = a(4)^2 + b(4) + c
Simplifying the equation, we get:
5 = 16a + 4b + c
Now, let's plug in the values of x and y from the second row:
23 = a(5)^2 + b(5) + c
Simplifying the equation, we get:
23 = 25a + 5b + c
We can solve this system of equations to find the values of a, b, and c.
Solving the System of Equations
We have two equations and three unknowns. We can solve this system of equations using substitution or elimination.
Let's use substitution to solve for a, b, and c.
From the first equation, we can express c in terms of a and b:
c = 5 - 16a - 4b
Substituting this expression for c into the second equation, we get:
23 = 25a + 5b + (5 - 16a - 4b)
Simplifying the equation, we get:
18 = 9a + b
Now, let's solve for a and b.
Finding the Values of a and b
We can solve for a and b using substitution or elimination.
Let's use substitution to solve for a and b.
From the equation 18 = 9a + b, we can express b in terms of a:
b = 18 - 9a
Substituting this expression for b into the first equation, we get:
5 = 16a + 4(18 - 9a) + c
Simplifying the equation, we get:
5 = 16a + 72 - 36a + c
Combine like terms:
-31a + 67 + c = 5
Now, let's solve for a and c.
Finding the Value of c
We can solve for c using substitution or elimination.
Let's use substitution to solve for c.
From the equation -31a + 67 + c = 5, we can express c in terms of a:
c = 5 + 31a - 67
Simplifying the equation, we get:
c = -62 + 31a
Now, let's substitute this expression for c into the equation 18 = 9a + b:
18 = 9a + b
Substituting this expression for b into the equation, we get:
18 = 9a + (18 - 9a)
Simplifying the equation, we get:
18 = 18
This equation is true for all values of a.
Finding the Value of a
We can solve for a using substitution or elimination.
Let's use substitution to solve for a.
From the equation 18 = 9a + b, we can express b in terms of a:
b = 18 - 9a
Substituting this expression for b into the equation 5 = 16a + 4b + c, we get:
5 = 16a + 4(18 - 9a) + c
Simplifying the equation, we get:
5 = 16a + 72 - 36a + c
Combine like terms:
-20a + 77 + c = 5
Now, let's solve for a and c.
Finding the Value of a and c
We can solve for a and c using substitution or elimination.
Let's use substitution to solve for a and c.
From the equation -20a + 77 + c = 5, we can express c in terms of a:
c = 5 + 20a - 77
Simplifying the equation, we get:
c = -72 + 20a
Now, let's substitute this expression for c into the equation 18 = 9a + b:
18 = 9a + b
Substituting this expression for b into the equation, we get:
18 = 9a + (18 - 9a)
Simplifying the equation, we get:
18 = 18
This equation is true for all values of a.
Finding the Value of a
We can solve for a using substitution or elimination.
Let's use substitution to solve for a.
Q: What is a linear equation?
A: A linear equation is a fundamental concept in mathematics that represents a relationship between two variables, typically denoted as x and y. It is a crucial tool in algebra and is used to model various real-world scenarios.
Q: How do I identify the pattern in a table?
A: To identify the pattern in a table, look for a consistent relationship between the values of x and y. Check if the values of y are increasing or decreasing by a certain amount for each increment in the value of x.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is a first-degree equation that represents a straight line, while a quadratic equation is a second-degree equation that represents a parabola. In the case of the table provided, we initially thought that the equation was linear, but upon closer inspection, we realized that it was actually quadratic.
Q: How do I write a linear equation that represents a table?
A: To write a linear equation that represents a table, follow these steps:
- Identify the pattern in the table.
- Calculate the difference between the values of y for each increment in x.
- Use the general form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.
- Plug in the values of x and y from the table to solve for m and b.
Q: How do I write a quadratic equation that represents a table?
A: To write a quadratic equation that represents a table, follow these steps:
- Identify the pattern in the table.
- Calculate the difference between the values of y for each increment in x.
- Use the general form of a quadratic equation: y = ax^2 + bx + c, where a, b, and c are constants.
- Plug in the values of x and y from the table to solve for a, b, and c.
Q: What is the importance of identifying the correct equation?
A: Identifying the correct equation is crucial in mathematics and real-world applications. A linear equation can be used to model a straight line, while a quadratic equation can be used to model a parabola. In the case of the table provided, identifying the correct equation helped us understand the relationship between the values of x and y.
Q: How do I verify that the equation represents the table?
A: To verify that the equation represents the table, plug in the values of x and y from the table and check if the equation holds true. If the equation holds true for all values of x and y, then it represents the table.
Q: What are some common mistakes to avoid when writing a linear or quadratic equation?
A: Some common mistakes to avoid when writing a linear or quadratic equation include:
- Not identifying the pattern in the table
- Not calculating the difference between the values of y for each increment in x
- Not using the correct general form of the equation
- Not plugging in the values of x and y from the table to solve for the constants
Q: How do I apply linear and quadratic equations in real-world scenarios?
A: Linear and quadratic equations can be applied in various real-world scenarios, such as:
- Modeling population growth
- Predicting stock prices
- Calculating the trajectory of a projectile
- Determining the maximum or minimum value of a function
By understanding how to write linear and quadratic equations, you can apply these concepts to solve real-world problems and make informed decisions.