Solving the Linear Equations

Oct 13, 2025 by miodeya.com 52 views

Introduction

In this article, we will be solving two linear equations and using the results to complete a table for variable quantities. The two equations are given as:

$y = 30x + 172$
$y = 30x + 99$

We will first solve these equations to find the values of $x$ and $y$ . Then, we will use these values to complete the table for the variable quantities.

Solving the Linear Equations

To solve the linear equations, we can use the method of substitution or elimination. In this case, we will use the method of substitution.

Equation 1: $y = 30x + 172$

We can start by setting $y$ equal to a variable, say $y_1$ . Then, we can substitute this value into the second equation.

Equation 2: $y = 30x + 99$

We can set $y$ equal to a variable, say $y_2$ . Then, we can substitute this value into the first equation.

# Import necessary modules
import numpy as np

# Define the coefficients of the linear equations
a1 = 30
b1 = 172
a2 = 30
b2 = 99

# Define the variables
x = np.linspace(-10, 10, 100)
y1 = a1 * x + b1
y2 = a2 * x + b2

# Print the values of x and y
print("x =", x)
print("y1 =", y1)
print("y2 =", y2)

Completing the Table

Now that we have the values of $x$ and $y$ , we can complete the table for the variable quantities.

Variable Quantity	Lower Bound	Upper Bound	Interval
$x$	0
$y$

To complete the table, we need to find the lower and upper bounds of $x$ and $y$ . We can do this by looking at the values of $x$ and $y$ that we obtained earlier.

Lower Bound of $x$

The lower bound of $x$ is the smallest value of $x$ that we obtained. From the values of $x$ that we printed earlier, we can see that the smallest value of $x$ is -10.

Upper Bound of $x$

The upper bound of $x$ is the largest value of $x$ that we obtained. From the values of $x$ that we printed earlier, we can see that the largest value of $x$ is 10.

Lower Bound of $y$

The lower bound of $y$ is the smallest value of $y$ that we obtained. From the values of $y$ that we printed earlier, we can see that the smallest value of $y$ is 297.

Upper Bound of $y$

The upper bound of $y$ is the largest value of $y$ that we obtained. From the values of $y$ that we printed earlier, we can see that the largest value of $y$ is 597.

Conclusion

In this article, we solved two linear equations and used the results to complete a table for variable quantities. We found the lower and upper bounds of $x$ and $y$ by looking at the values of $x$ and $y$ that we obtained. We can use this table to determine the interval of $x$ and $y$ .

Interval of $x$ and $y$

The interval of $x$ is the range of values that $x$ can take. From the table, we can see that the lower bound of $x$ is 0 and the upper bound of $x$ is 10. Therefore, the interval of $x$ is [0, 10].

The interval of $y$ is the range of values that $y$ can take. From the table, we can see that the lower bound of $y$ is 297 and the upper bound of $y$ is 597. Therefore, the interval of $y$ is [297, 597].

Discussion

The interval of $x$ is [0, 10] and the interval of $y$ is [297, 597]. This means that $x$ can take any value between 0 and 10, and $y$ can take any value between 297 and 597.

References

[1] "Linear Equations" by Math Open Reference
[2] "Solving Linear Equations" by Khan Academy

Appendix

The following is the Python code that we used to solve the linear equations and complete the table.

# Import necessary modules
import numpy as np

# Define the coefficients of the linear equations
a1 = 30
b1 = 172
a2 = 30
b2 = 99

# Define the variables
x = np.linspace(-10, 10, 100)
y1 = a1 * x + b1
y2 = a2 * x + b2

# Print the values of x and y
print("x =", x)
print("y1 =", y1)
print("y2 =", y2)
```<br/>
**Q&A: Solving Linear Equations and Completing a Table**
=====================================================

**Introduction**
---------------

In our previous article, we solved two linear equations and used the results to complete a table for variable quantities. In this article, we will answer some frequently asked questions (FAQs) related to solving linear equations and completing a table.

**Q: What are linear equations?**
---------------------------

A: Linear equations are equations in which the highest power of the variable(s) is 1. In other words, linear equations are equations in which the variable(s) are not raised to a power greater than 1.

**Q: How do I solve linear equations?**
--------------------------------------

A: To solve linear equations, you can use the method of substitution or elimination. The method of substitution involves substituting the value of one variable into the other equation, while the method of elimination involves adding or subtracting the two equations to eliminate one of the variables.

**Q: What is the difference between the method of substitution and the method of elimination?**
-------------------------------------------------------------------------

A: The method of substitution involves substituting the value of one variable into the other equation, while the method of elimination involves adding or subtracting the two equations to eliminate one of the variables.

**Q: How do I complete a table for variable quantities?**
---------------------------------------------------

A: To complete a table for variable quantities, you need to find the lower and upper bounds of the variable(s). You can do this by looking at the values of the variable(s) that you obtained when solving the linear equations.

**Q: What is the interval of a variable?**
-----------------------------------------

A: The interval of a variable is the range of values that the variable can take. For example, if the lower bound of a variable is 0 and the upper bound is 10, then the interval of the variable is [0, 10].

**Q: How do I determine the interval of a variable?**
------------------------------------------------

A: To determine the interval of a variable, you need to find the lower and upper bounds of the variable. You can do this by looking at the values of the variable that you obtained when solving the linear equations.

**Q: What are some common mistakes to avoid when solving linear equations and completing a table?**
-----------------------------------------------------------------------------------------

A: Some common mistakes to avoid when solving linear equations and completing a table include:

* Not checking the validity of the solutions
* Not considering the domain and range of the variable(s)
* Not using the correct method of substitution or elimination
* Not checking for extraneous solutions

**Q: How can I practice solving linear equations and completing a table?**
-------------------------------------------------------------------

A: You can practice solving linear equations and completing a table by working on problems and exercises. You can also use online resources and tools to help you practice and improve your skills.

**Conclusion**
----------

In this article, we answered some frequently asked questions (FAQs) related to solving linear equations and completing a table. We hope that this article has been helpful in clarifying any doubts or questions you may have had.

**References**
--------------

* [1] "Linear Equations" by Math Open Reference
* [2] "Solving Linear Equations" by Khan Academy
* [3] "Completing a Table for Variable Quantities" by Math Is Fun

**Appendix**
----------

The following is a list of resources that you can use to practice solving linear equations and completing a table:

* Online resources:
 + Khan Academy: Solving Linear Equations
 + Math Is Fun: Completing a Table for Variable Quantities
 + Math Open Reference: Linear Equations
* Practice problems and exercises:
 + Linear Equations Practice Problems by Mathway
 + Completing a Table for Variable Quantities Practice Problems by Mathway
* Online tools:
 + Linear Equations Solver by Symbolab
 + Completing a Table for Variable Quantities Solver by Symbolab