Solve For { X $}$. ${ 5x + 7x = 72 }$

Mar 8, 2025 by ADMIN 39 views

Solving Linear Equations: A Step-by-Step Guide to Finding the Value of x

Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving linear equations of the form ax + b = c, where a, b, and c are constants.

Understanding the Given Equation

The given equation is 5x + 7x = 72. This equation is a linear equation in one variable, x. Our goal is to solve for the value of x.

Combining Like Terms

To solve the equation, we need to combine like terms. In this case, the like terms are the terms with the variable x. We can combine the two terms with x by adding their coefficients.

# Import necessary modules
import sympy as sp
x = sp.symbols('x')

equation = 5x + 7x - 72

solution = sp.solve(equation, x)

Simplifying the Equation

By combining the like terms, we get 12x = 72. This is a simplified form of the original equation.

Isolating the Variable

To isolate the variable x, we need to get rid of the coefficient 12. We can do this by dividing both sides of the equation by 12.

# Divide both sides of the equation by 12
solution = sp.solve(12*x - 72, x)

Solving for x

By dividing both sides of the equation by 12, we get x = 6. This is the solution to the equation.

Conclusion

In this article, we solved a linear equation of the form ax + b = c. We combined like terms, simplified the equation, and isolated the variable x. By following these steps, we were able to find the value of x, which is 6.

Tips and Tricks

When solving linear equations, always combine like terms first.
Use the distributive property to simplify the equation.
Isolate the variable by dividing both sides of the equation by the coefficient.
Check your solution by plugging it back into the original equation.

Real-World Applications

Linear equations have numerous real-world applications. For example, in physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the behavior of markets. In engineering, linear equations are used to design and optimize systems.

Common Mistakes

Failing to combine like terms.
Not using the distributive property to simplify the equation.
Not isolating the variable by dividing both sides of the equation by the coefficient.
Not checking the solution by plugging it back into the original equation.

Final Thoughts

Solving linear equations is a fundamental skill that is essential in various fields. By following the steps outlined in this article, you can solve linear equations with ease. Remember to combine like terms, simplify the equation, and isolate the variable. With practice and patience, you will become proficient in solving linear equations.

Additional Resources

Khan Academy: Linear Equations
Mathway: Linear Equations
Wolfram Alpha: Linear Equations

Frequently Asked Questions

Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable(s) is 1.
Q: How do I solve a linear equation? A: To solve a linear equation, combine like terms, simplify the equation, and isolate the variable.
Q: What is the distributive property? A: The distributive property is a mathematical property that states that a(b + c) = ab + ac.
Q: How do I check my solution? A: To check your solution, plug it back into the original equation and verify that it is true.

Introduction

Solving linear equations is a fundamental skill that is essential in various fields such as physics, engineering, and economics. In our previous article, we provided a step-by-step guide on how to solve linear equations. In this article, we will answer some of the most frequently asked questions about solving linear equations.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. For example, 2x + 3 = 5 is a linear equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, combine like terms, simplify the equation, and isolate the variable. For example, to solve the equation 2x + 3 = 5, we can combine the like terms, simplify the equation, and isolate the variable x.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a(b + c) = ab + ac. For example, 2(x + 3) = 2x + 6.

Q: How do I check my solution?

A: To check your solution, plug it back into the original equation and verify that it is true. For example, if we solve the equation 2x + 3 = 5 and get x = 1, we can plug x = 1 back into the original equation to verify that it is true.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 2x + 1 = 0 is a quadratic equation.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, we can use the method of substitution or the method of elimination. For example, if we have the system of equations x + y = 2 and x - y = 1, we can use the method of substitution to solve for x and y.

Q: What is the significance of linear equations in real-world applications?

A: Linear equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, linear equations are used to describe the motion of objects, model the behavior of markets, and design and optimize systems.

Q: How do I graph a linear equation?

A: To graph a linear equation, we can use the slope-intercept form of the equation, which is y = mx + b, where m is the slope and b is the y-intercept. For example, the equation y = 2x + 3 can be graphed by plotting the points (0, 3) and (1, 5).

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include failing to combine like terms, not using the distributive property to simplify the equation, and not isolating the variable by dividing both sides of the equation by the coefficient.