What Is The Solution To This Equation?$\[ 3(4x + 6) = 9x + 12 \\]A. \[$ X = -10 \$\] B. \[$ X = 10 \$\] C. \[$ X = -2 \$\] D. \[$ X = 2 \$\]
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 linear equation, 3(4x + 6) = 9x + 12, and explore the different solution options provided.
Understanding the Equation
The given equation is 3(4x + 6) = 9x + 12. To solve this equation, we need to isolate the variable x. The equation involves parentheses, which need to be evaluated first.
Step 1: Evaluate the Parentheses
The first step is to evaluate the expression inside the parentheses: 4x + 6. This expression is multiplied by 3, so we need to multiply each term inside the parentheses by 3.
3(4x + 6) = 3(4x) + 3(6) = 12x + 18
Step 2: Simplify the Equation
Now that we have evaluated the parentheses, we can simplify the equation by combining like terms.
12x + 18 = 9x + 12
Step 3: Isolate the Variable
To isolate the variable x, we need to get all the terms involving x on one side of the equation and the constant terms on the other side. We can do this by subtracting 9x from both sides of the equation.
12x - 9x + 18 = 9x - 9x + 12 3x + 18 = 12
Step 4: Solve for x
Now that we have isolated the variable x, we can solve for its value. To do this, we need to get rid of the constant term on the same side as x. We can do this by subtracting 18 from both sides of the equation.
3x + 18 - 18 = 12 - 18 3x = -6
Step 5: Find the Value of x
Finally, we can find the value of x by dividing both sides of the equation by 3.
3x / 3 = -6 / 3 x = -2
Conclusion
In conclusion, the solution to the equation 3(4x + 6) = 9x + 12 is x = -2. This is the correct answer among the options provided.
Why is x = -2 the Correct Answer?
To understand why x = -2 is the correct answer, let's substitute x = -2 into the original equation and see if it holds true.
3(4(-2) + 6) = 9(-2) + 12 3(-8 + 6) = -18 + 12 3(-2) = -6 -6 = -6
As we can see, the equation holds true when x = -2. This confirms that x = -2 is indeed the correct solution.
What is the Importance of Solving Linear Equations?
Solving linear equations is an essential skill in mathematics, and it has numerous applications in real-life situations. Some of the importance of solving linear equations include:
- Problem-solving: Solving linear equations helps us to solve problems in various fields, such as physics, engineering, economics, and computer science.
- Critical thinking: Solving linear equations requires critical thinking and analytical skills, which are essential for making informed decisions in personal and professional life.
- Mathematical modeling: Solving linear equations helps us to create mathematical models that can be used to describe and analyze real-world phenomena.
- Data analysis: Solving linear equations is an essential tool for data analysis, as it helps us to extract insights and patterns from data.
Conclusion
In conclusion, solving linear equations is a crucial skill that has numerous applications in real-life situations. By following the steps outlined in this article, we can solve linear equations and extract insights and patterns from data. Whether you are a student or a professional, mastering the art of solving linear equations is essential for success in various fields.
Final Thoughts
Solving linear equations is a fundamental concept in mathematics, and it has numerous applications in real-life situations. By following the steps outlined in this article, we can solve linear equations and extract insights and patterns from data. Whether you are a student or a professional, mastering the art of solving linear equations is essential for success in various fields.
References
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: 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, whereas 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 + 4x + 4 = 0 is a quadratic equation.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. For example, to solve the equation 2x + 3 = 5, you can subtract 3 from both sides to get 2x = 2, and then divide both sides by 2 to get x = 1.
Q: What is the order of operations when solving a linear equation?
A: When solving a linear equation, you need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate any expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I handle fractions when solving a linear equation?
A: When solving a linear equation that involves fractions, you can multiply both sides of the equation by the denominator of the fraction to eliminate the fraction. For example, to solve the equation x/2 = 3, you can multiply both sides by 2 to get x = 6.
Q: What is the difference between a linear equation and a system of linear equations?
A: A linear equation is a single equation with one variable, whereas a system of linear equations is a set of two or more linear equations with the same variable(s). For example, the equation 2x + 3 = 5 is a linear equation, while the system of equations {2x + 3 = 5, x - 2 = 1} is a system of linear equations.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you can use various methods, such as substitution, elimination, or graphing. For example, to solve the system of equations {2x + 3 = 5, x - 2 = 1}, you can use the substitution method by solving one equation for one variable and then substituting that expression into the other equation.
Q: What is the importance of solving linear equations?
A: Solving linear equations is an essential skill in mathematics and has numerous applications in real-life situations, such as physics, engineering, economics, and computer science. It helps us to solve problems, make informed decisions, and analyze data.
Q: How can I practice solving linear equations?
A: You can practice solving linear equations by working on exercises and problems from textbooks, online resources, or worksheets. You can also try solving real-world problems that involve linear equations, such as calculating the cost of goods or determining the time it takes to complete a task.
Conclusion
In conclusion, solving linear equations is a crucial skill that has numerous applications in real-life situations. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations and apply this skill to various fields. Whether you are a student or a professional, mastering the art of solving linear equations is essential for success in various fields.