Write The Value For { X $}$ That Makes The Equation { 1 - 2x + 5 = 4x - 3 $}$ True.

Mar 12, 2025 by ADMIN 84 views

Solving Linear Equations: Finding the Value of x

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, ${$ 1 - 2x + 5 = 4x - 3 $}$, to find the value of x that makes the equation true. We will break down the solution step by step, using algebraic manipulations to isolate the variable x.

Understanding the Equation

The given equation is a linear equation in one variable, x. It is a statement that two expressions are equal, and our goal is to find the value of x that makes this statement true. The equation is:

${$ 1 - 2x + 5 = 4x - 3 $}$

Step 1: Simplify the Equation

To solve the equation, we first need to simplify it by combining like terms. We can start by combining the constants on the left-hand side:

${$ 1 + 5 - 2x = 4x - 3 $}$

This simplifies to:

${$ 6 - 2x = 4x - 3 $}$

Step 2: Isolate the Variable x

Next, we need to isolate the variable x on one side of the equation. We can do this by adding 2x to both sides of the equation:

${$ 6 = 6x - 3 $}$

Step 3: Add 3 to Both Sides

To get rid of the negative term on the right-hand side, we can add 3 to both sides of the equation:

${$ 9 = 6x $}$

Step 4: Divide Both Sides by 6

Finally, we can solve for x by dividing both sides of the equation by 6:

${$ x = \frac{9}{6} $}$

Step 5: Simplify the Fraction

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

${$ x = \frac{3}{2} $}$

Conclusion

In this article, we solved a linear equation to find the value of x that makes the equation true. We started by simplifying the equation, then isolated the variable x, and finally solved for x by dividing both sides of the equation by 6. The value of x that makes the equation true is ${$ \frac{3}{2} $}$.

Applications of Linear Equations

Linear equations have many real-world applications, including:

Physics and Engineering: Linear equations are used to model the motion of objects, the flow of fluids, and the behavior of electrical circuits.
Economics: Linear equations are used to model the behavior of economic systems, including the supply and demand of goods and services.
Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Tips for Solving Linear Equations

Here are some tips for solving linear equations:

Use algebraic manipulations: Linear equations can be solved using algebraic manipulations, such as adding, subtracting, multiplying, and dividing both sides of the equation.
Isolate the variable: To solve for x, isolate the variable x on one side of the equation.
Check your work: Once you have solved for x, check your work by plugging the value of x back into the original equation.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear equations:

Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
Not isolating the variable: Failing to isolate the variable x can make it difficult to solve for x.
Not checking your work: Failing to check your work can lead to incorrect solutions.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear equations and find the value of x that makes the equation true. Remember to simplify the equation, isolate the variable x, and check your work to ensure that your solution is correct. With practice and patience, you can become proficient in solving linear equations and apply this skill to real-world problems.

Introduction

In our previous article, we solved a linear equation to find the value of x that makes the equation true. In this article, we will answer some frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. It is a statement that two expressions are equal, and our goal is to find the value of x that makes this statement true.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, combine like terms on both sides of the equation. This involves adding or subtracting the coefficients of the same variable.

Q: How do I isolate the variable x?

A: To isolate the variable x, add or subtract the same value to both sides of the equation to get rid of the constant term. Then, divide both sides of the equation by the coefficient of x to solve for x.

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
Not isolating the variable x
Not checking your work
Dividing by zero
Not following the order of operations

Q: How do I check my work?

A: To check your work, plug the value of x back into the original equation and simplify. If the equation is true, then your solution is correct.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

Physics and Engineering: Linear equations are used to model the motion of objects, the flow of fluids, and the behavior of electrical circuits.
Economics: Linear equations are used to model the behavior of economic systems, including the supply and demand of goods and services.
Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: How do I solve a linear equation with decimals?

A: To solve a linear equation with decimals, multiply both sides of the equation by 10 to eliminate the decimals.

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 (x) is 1, while a quadratic equation is an equation in which the highest power of the variable (x) is 2.

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

A: To solve a system of linear equations, use the method of substitution or elimination to find the values of the variables.

Conclusion

Additional Resources

For more information on solving linear equations, check out the following resources:

Mathway: A online math problem solver that can help you solve linear equations.
Khan Academy: A free online resource that provides video lessons and practice exercises on solving linear equations.
Math Open Reference: A free online reference book that provides detailed explanations and examples of solving linear equations.

Final Thoughts

Solving linear equations is a fundamental skill that is used in many areas of mathematics and science. By mastering this skill, you can solve a wide range of problems and apply this skill to real-world situations. Remember to practice regularly and seek help when you need it. With time and effort, you can become proficient in solving linear equations and achieve your goals.