Solve The Equation For { X$} : : : { 5 - 2(3 - X) = 4x + 10 \}
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, 5 - 2(3 - x) = 4x + 10, and provide a step-by-step guide on how to solve for x. We will also discuss the importance of linear equations in real-life applications and provide tips on how to approach solving linear equations.
What are Linear Equations?
Linear equations are algebraic equations in which the highest power of the variable(s) is 1. They can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.
The Equation to be Solved
The equation we will be solving is 5 - 2(3 - x) = 4x + 10. This equation is a linear equation, and we will use algebraic manipulation to solve for x.
Step 1: Distribute the Negative 2
To solve the equation, we first need to distribute the negative 2 to the terms inside the parentheses. This will give us:
5 - 2(3 - x) = 5 - 6 + 2x
Step 2: Simplify the Equation
Next, we simplify the equation by combining like terms. This will give us:
-1 + 2x = 4x + 10
Step 3: Add 1 to Both Sides
To isolate the term with x, we add 1 to both sides of the equation. This will give us:
2x = 4x + 11
Step 4: Subtract 4x from Both Sides
Next, we subtract 4x from both sides of the equation to get:
-2x = 11
Step 5: Divide Both Sides by -2
Finally, we divide both sides of the equation by -2 to solve for x. This will give us:
x = -11/2
Conclusion
In this article, we solved the linear equation 5 - 2(3 - x) = 4x + 10 using algebraic manipulation. We distributed the negative 2, simplified the equation, added 1 to both sides, subtracted 4x from both sides, and finally divided both sides by -2 to solve for x. The solution to the equation is x = -11/2.
Tips for Solving Linear Equations
Here are some tips for solving linear equations:
- Always start by simplifying the equation by combining like terms.
- Use inverse operations to isolate the term with x.
- Be careful when adding or subtracting fractions.
- Check your solution by plugging it back into the original equation.
Real-Life Applications of Linear Equations
Linear equations have many real-life applications, including:
- Physics: Linear equations are used to describe the motion of objects under constant acceleration.
- Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: Linear equations are used to model economic systems and make predictions about future trends.
Conclusion
Solving linear equations is a crucial skill for students to master. In this article, we solved the linear equation 5 - 2(3 - x) = 4x + 10 using algebraic manipulation. We provided a step-by-step guide on how to solve for x and discussed the importance of linear equations in real-life applications. We also provided tips on how to approach solving linear equations and highlighted the many real-life applications of linear equations.
Frequently Asked Questions
- Q: What is a linear equation? A: A linear equation is an algebraic 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, start by simplifying the equation by combining like terms, use inverse operations to isolate the term with x, and check your solution by plugging it back into the original equation.
- Q: What are some real-life applications of linear equations? A: Linear equations have many real-life applications, including physics, engineering, and economics.
References
- [1] "Algebra" by Michael Artin
- [2] "Linear Algebra and Its Applications" by Gilbert Strang
- [3] "Mathematics for Engineers and Scientists" by Donald R. Hill
Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.
Introduction
Solving linear equations is a crucial skill for students to master. In our previous article, we provided a step-by-step guide on how to solve the linear equation 5 - 2(3 - x) = 4x + 10. In this article, we will provide a Q&A guide to help students understand and apply the concepts of linear equations.
Q&A Guide
Q: What is a linear equation?
A: A linear equation is an algebraic equation in which the highest power of the variable(s) is 1. It can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.
Q: How do I solve a linear equation?
A: To solve a linear equation, start by simplifying the equation by combining like terms, use inverse operations to isolate the term with x, and check your solution by plugging it back into the original equation.
Q: What are some common types of linear equations?
A: Some common types of linear equations include:
- Simple linear equations: Equations in the form ax + b = c, where a, b, and c are constants.
- Linear equations with fractions: Equations in the form ax/b + c = d, where a, b, c, and d are constants.
- Linear equations with decimals: Equations in the form ax + b = c, where a, b, and c are decimals.
Q: How do I simplify a linear equation?
A: To simplify a linear equation, combine like terms by adding or subtracting the coefficients of the same variable.
Q: What is the inverse operation of addition?
A: The inverse operation of addition is subtraction.
Q: What is the inverse operation of multiplication?
A: The inverse operation of multiplication is division.
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.
Q: What are some real-life applications of linear equations?
A: Linear equations have many real-life applications, including:
- Physics: Linear equations are used to describe the motion of objects under constant acceleration.
- Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: Linear equations are used to model economic systems and make predictions about future trends.
Q: Can I use a calculator to solve linear equations?
A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your solution by plugging it back into the original equation.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not simplifying the equation: Make sure to simplify the equation by combining like terms.
- Not using inverse operations: Use inverse operations to isolate the term with x.
- Not checking the solution: Always check your solution by plugging it back into the original equation.
Conclusion
Solving linear equations is a crucial skill for students to master. In this article, we provided a Q&A guide to help students understand and apply the concepts of linear equations. We covered common types of linear equations, how to simplify and solve linear equations, and how to check solutions. We also discussed real-life applications of linear equations and common mistakes to avoid.
Frequently Asked Questions
- Q: What is a linear equation? A: A linear equation is an algebraic 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, start by simplifying the equation by combining like terms, use inverse operations to isolate the term with x, and check your solution by plugging it back into the original equation.
- Q: What are some real-life applications of linear equations? A: Linear equations have many real-life applications, including physics, engineering, and economics.
References
- [1] "Algebra" by Michael Artin
- [2] "Linear Algebra and Its Applications" by Gilbert Strang
- [3] "Mathematics for Engineers and Scientists" by Donald R. Hill
Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.