Solve The Equation For \[$ X \$\].a) \[$\frac{4x - 1}{2} = X + 7\$\]
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, which is a first-degree equation in one variable. We will use the equation {\frac{4x - 1}{2} = x + 7$}$ as an example to demonstrate the step-by-step process of solving linear equations.
What are Linear Equations?
A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that 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 {\frac{4x - 1}{2} = x + 7$}$. This equation is a linear equation in one variable, and it can be solved using algebraic manipulation.
Step 1: Multiply Both Sides by 2
To eliminate the fraction, we can multiply both sides of the equation by 2. This will give us:
${$4x - 1 = 2(x + 7)$]
Step 2: Distribute the 2 on the Right-Hand Side
Next, we can distribute the 2 on the right-hand side of the equation. This will give us:
[$4x - 1 = 2x + 14$]
Step 3: Subtract 2x from Both Sides
To isolate the variable [x\$}, we can subtract ${2x\$} from both sides of the equation. This will give us:
${$2x - 1 = 14$]
Step 4: Add 1 to Both Sides
Next, we can add 1 to both sides of the equation. This will give us:
[$2x = 15$]
Step 5: Divide Both Sides by 2
Finally, we can divide both sides of the equation by 2. This will give us:
[$x = \frac{15}{2}$]
Conclusion
Solving linear equations is an essential skill for students and professionals alike. By following the step-by-step process outlined in this article, we can solve linear equations with ease. Remember to multiply both sides of the equation by the same value to eliminate fractions, distribute the value on the right-hand side, subtract the variable from both sides, add the constant to both sides, and finally divide both sides by the coefficient of the variable.
Example Problems
Problem 1
Solve the equation [\frac{3x + 2}{4} = x - 3\$}.
Step 1: Multiply Both Sides by 4
To eliminate the fraction, we can multiply both sides of the equation by 4. This will give us:
${$3x + 2 = 4(x - 3)$]
Step 2: Distribute the 4 on the Right-Hand Side
Next, we can distribute the 4 on the right-hand side of the equation. This will give us:
[$3x + 2 = 4x - 12$]
Step 3: Subtract 3x from Both Sides
To isolate the variable [x\$}, we can subtract ${3x\$} from both sides of the equation. This will give us:
${$2 = x - 12$]
Step 4: Add 12 to Both Sides
Next, we can add 12 to both sides of the equation. This will give us:
[$14 = x$]
Step 5: Write the Solution
The solution to the equation is [x = 14\$}.
Problem 2
Solve the equation {\frac{2x - 5}{3} = x + 2$}$.
Step 1: Multiply Both Sides by 3
To eliminate the fraction, we can multiply both sides of the equation by 3. This will give us:
${$2x - 5 = 3(x + 2)$]
Step 2: Distribute the 3 on the Right-Hand Side
Next, we can distribute the 3 on the right-hand side of the equation. This will give us:
[$2x - 5 = 3x + 6$]
Step 3: Subtract 2x from Both Sides
To isolate the variable [x\$}, we can subtract ${2x\$} from both sides of the equation. This will give us:
{-5 = x + 6$]
Step 4: Subtract 6 from Both Sides
Next, we can subtract 6 from both sides of the equation. This will give us:
[$-11 = x$]
Step 5: Write the Solution
The solution to the equation is [x = -11\$}.
Tips and Tricks
- Always multiply both sides of the equation by the same value to eliminate fractions.
- Distribute the value on the right-hand side of the equation.
- Subtract the variable from both sides of the equation.
- Add the constant to both sides of the equation.
- Finally, divide both sides of the equation by the coefficient of the variable.
Introduction
Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. In this article, we will provide a Q&A guide to help you understand the process of solving linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that 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, you need to follow these steps:
- Multiply both sides of the equation by the same value to eliminate fractions.
- Distribute the value on the right-hand side of the equation.
- Subtract the variable from both sides of the equation.
- Add the constant to both sides of the equation.
- Finally, divide both sides of the equation by the coefficient of the variable.
Q: What is the coefficient of the variable?
A: The coefficient of the variable is the number that is multiplied by the variable. For example, in the equation ${2x = 5\$}, the coefficient of the variable {x$}$ is 2.
Q: How do I eliminate fractions in a linear equation?
A: To eliminate fractions in a linear equation, you need to multiply both sides of the equation by the same value. This will get rid of the fraction and make it easier to solve the equation.
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, the equation {x + 2 = 3$}$ is a linear equation, while the equation {x^2 + 2x + 1 = 0$}$ is a quadratic equation.
Q: Can I use a calculator to solve linear equations?
A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the steps involved in solving the equation, as using a calculator without understanding the process can lead to errors.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not multiplying both sides of the equation by the same value to eliminate fractions.
- Not distributing the value on the right-hand side of the equation.
- Not subtracting the variable from both sides of the equation.
- Not adding the constant to both sides of the equation.
- Not dividing both sides of the equation by the coefficient of the variable.
Q: How can I practice solving linear equations?
A: You can practice solving linear equations by:
- Working through example problems.
- Using online resources, such as math websites and apps.
- Asking a teacher or tutor for help.
- Joining a study group or math club.
By following these tips and practicing regularly, you can become proficient in solving linear equations.
Conclusion
Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations. Remember to avoid common mistakes, such as not multiplying both sides of the equation by the same value, and to use a calculator only when necessary.