Solve The Equation: { -3x + 1 + 10x = X + 4$}$Choose The Correct Solution:A. { X = \frac{1}{2}$}$ B. { X = \frac{5}{6}$}$ C. { X = 12$}$ D. { X = 18$}$
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, step by step, and provide a clear explanation of the process. We will also discuss the importance of linear equations in real-life applications and provide examples of how they are used in various fields.
What is a Linear Equation?
A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It is a simple equation that can be solved using basic algebraic operations. Linear equations can be written in the form ax + b = c, where a, b, and c are constants.
The Equation to be Solved
The equation we will be solving is:
-3x + 1 + 10x = x + 4
Step 1: Simplify the Equation
To simplify the equation, we need to combine like terms. In this case, we have two terms with x: -3x and 10x. We can combine these terms by adding their coefficients (the numbers in front of the x).
-3x + 10x = 7x
So, the simplified equation is:
7x + 1 = x + 4
Step 2: Isolate the Variable
To isolate the variable x, we need to get all the terms with x on one side of the equation. We can do this by subtracting x from both sides of the equation.
7x - x = 4 - 1
This simplifies to:
6x = 3
Step 3: Solve for x
To solve for x, we need to isolate x by dividing both sides of the equation by 6.
x = 3/6
x = 1/2
Conclusion
We have successfully solved the linear equation -3x + 1 + 10x = x + 4. The solution is x = 1/2. This equation is a simple example of a linear equation, and solving it requires basic algebraic operations.
Real-Life Applications of Linear Equations
Linear equations have numerous real-life applications in various fields, including:
- Physics: Linear equations are used to describe the motion of objects, such as the trajectory of a projectile or the velocity of a car.
- Engineering: Linear equations are used to design and optimize systems, such as electrical circuits or mechanical systems.
- Economics: Linear equations are used to model economic systems, such as supply and demand curves or cost-benefit analysis.
- Computer Science: Linear equations are used in algorithms and data structures, such as linear search or sorting algorithms.
Examples of Linear Equations in Real-Life Scenarios
- Traffic Flow: A linear equation can be used to model the flow of traffic on a highway. For example, if the speed of traffic is 60 km/h and the distance between two points is 10 km, the time it takes to travel between the two points can be calculated using a linear equation.
- Cost-Benefit Analysis: A linear equation can be used to calculate the cost of a project versus its benefits. For example, if the cost of a project is $100,000 and the benefits are $150,000, a linear equation can be used to determine the net benefit of the project.
- Supply and Demand: A linear equation can be used to model the supply and demand of a product. For example, if the supply of a product is 100 units and the demand is 120 units, a linear equation can be used to determine the price of the product.
Conclusion
In conclusion, linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. The equation -3x + 1 + 10x = x + 4 is a simple example of a linear equation, and solving it requires basic algebraic operations. Linear equations have numerous real-life applications in various fields, including physics, engineering, economics, and computer science. By understanding and solving linear equations, students can develop problem-solving skills and apply mathematical concepts to real-life scenarios.
Solutions
The correct solution to the equation -3x + 1 + 10x = x + 4 is:
x = 1/2
This is option A.
References
- Algebra: A Comprehensive Introduction by Michael Artin
- Linear Algebra: A Modern Introduction by David Poole
- Mathematics for Computer Science: By Eric Lehman, F Thomson Leighton, and Albert R Meyer
Additional Resources
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- Wolfram Alpha: Linear Equations
Frequently Asked Questions (FAQs) about Linear Equations =============================================================
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It is a simple equation that can be solved using basic algebraic operations.
Q: How do I simplify a linear equation?
A: To simplify a linear equation, you need to combine like terms. This involves adding or subtracting the coefficients (the numbers in front of the variable) of the terms with the same variable.
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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable by performing the necessary operations to get all the terms with the variable on one side of the equation.
Q: What are some real-life applications of linear equations?
A: Linear equations have numerous real-life applications in various fields, including physics, engineering, economics, and computer science.
Q: Can you give me an example of a linear equation in real-life?
A: Yes, here's an example: If you have a budget of $100 to spend on a trip, and you want to know how many days you can stay in a hotel that costs $20 per night, you can use a linear equation to solve the problem.
Q: How do I determine if an equation is linear or not?
A: To determine if an equation is linear or not, you need to check if the highest power of the variable is 1. If it is, then the equation is linear.
Q: Can you explain the concept of slope in linear equations?
A: Yes, the slope of a linear equation is a measure of how steep the line is. It is calculated by dividing the change in the y-coordinate by the change in the x-coordinate.
Q: How do I graph a linear equation?
A: To graph a linear equation, you need to plot two points on the graph and then draw a line through them.
Q: Can you give me some tips for solving linear equations?
A: Yes, here are some tips:
- Make sure to simplify the equation before solving it.
- Use the correct order of operations (PEMDAS).
- Check your work by plugging the solution 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 before solving it.
- Using the wrong order of operations.
- Not checking the solution by plugging it back into the original equation.
Q: Can you explain the concept of linear equations in three variables?
A: Yes, a linear equation in three variables is an equation in which the highest power of each variable is 1. It is a more complex equation than a linear equation in two variables.
Q: How do I solve a linear equation in three variables?
A: To solve a linear equation in three variables, you need to isolate one of the variables by performing the necessary operations to get all the terms with that variable on one side of the equation.
Q: Can you give me some examples of linear equations in three variables?
A: Yes, here are some examples:
- 2x + 3y - z = 4
- x - 2y + 3z = 5
- 3x + 2y - z = 6
Q: What are some real-life applications of linear equations in three variables?
A: Linear equations in three variables have numerous real-life applications in various fields, including physics, engineering, economics, and computer science.
Conclusion
In conclusion, linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. By understanding and solving linear equations, students can develop problem-solving skills and apply mathematical concepts to real-life scenarios.