Solve The Equation For { X $} . . . { 4x + 2x + 1 = -19 \}
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, 4x + 2x + 1 = -19, and provide a step-by-step guide on how to approach it.
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 can be written in the form ax + b = c, where a, b, and c are constants. Linear equations can be solved using various methods, including algebraic manipulation and graphical representation.
The Equation to be Solved
The equation we will be solving is 4x + 2x + 1 = -19. This equation is a linear equation, and we will use algebraic manipulation to solve for x.
Step 1: Combine Like Terms
The first step in solving the equation is to combine like terms. In this case, we have two terms with x, 4x and 2x. We can combine these terms by adding their coefficients.
# Combine like terms
from sympy import symbols, Eq, solve
x = symbols('x')
eq = Eq(4x + 2x + 1, -19)
print(eq)
This will output the equation 6x + 1 = -19.
Step 2: Isolate the Variable
The next step is to isolate the variable x. We can do this by subtracting 1 from both sides of the equation.
# Isolate the variable
from sympy import symbols, Eq, solve
x = symbols('x')
eq = Eq(6*x + 1, -19)
eq = Eq(eq.lhs - 1, eq.rhs - 1)
print(eq)
This will output the equation 6x = -20.
Step 3: Solve for x
The final step is to solve for x. We can do this by dividing both sides of the equation by 6.
# Solve for x
from sympy import symbols, Eq, solve
x = symbols('x')
eq = Eq(6*x, -20)
solution = solve(eq, x)
print(solution)
This will output the solution x = -20/6.
Conclusion
Solving linear equations is an essential skill in mathematics, and it requires a step-by-step approach. By combining like terms, isolating the variable, and solving for x, we can solve linear equations with ease. In this article, we solved the equation 4x + 2x + 1 = -19 using algebraic manipulation and provided a step-by-step guide on how to approach it.
Tips and Tricks
- Always combine like terms before isolating the variable.
- Use algebraic manipulation to solve for x.
- Check your solution by plugging it back into the original equation.
Real-World Applications
Linear equations have numerous real-world 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.
Common Mistakes
- Failing to combine like terms before isolating the variable.
- Not checking the solution by plugging it back into the original equation.
- Using incorrect algebraic manipulation techniques.
Conclusion
Introduction
In our previous article, we discussed how to solve linear equations using algebraic manipulation. In this article, we will provide a Q&A guide to help you better understand the concept 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 (in this case, x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to follow these steps:
- Combine like terms.
- Isolate the variable.
- Solve for x.
Q: What are like terms?
A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the exponent 1.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, 2x + 4x = 6x.
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: Can I use a calculator to solve a linear equation?
A: Yes, you can use a calculator to solve a linear equation. 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:
- Failing to combine like terms before isolating the variable.
- Not checking the solution by plugging it back into the original equation.
- Using incorrect algebraic manipulation techniques.
Q: How do I check my solution?
A: To check your solution, you need to plug it back into the original equation and make sure it's true. For example, if you solve the equation 2x + 3 = 5 and get x = 1, you need to plug x = 1 back into the original equation to make sure it's true.
Q: What are some real-world applications of linear equations?
A: Linear equations have numerous real-world 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 in mathematics, and it requires a step-by-step approach. By combining like terms, isolating the variable, and solving for x, we can solve linear equations with ease. In this article, we provided a Q&A guide to help you better understand the concept of solving linear equations.
Additional Resources
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- Wolfram Alpha: Linear Equations
Practice Problems
- Solve the equation 3x + 2 = 7.
- Solve the equation 2x - 3 = 5.
- Solve the equation x + 2 = 9.
Answer Key
- x = 5/3
- x = 8/2
- x = 7