Solve:4x-2\2x+3=5x+1\4x+5
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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 involves multiple equations with variables. We will use the given equation 4x - 2 / 2x + 3 = 5x + 1 / 4x + 5 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, substitution, and elimination.
The Given Equation
The given equation is 4x - 2 / 2x + 3 = 5x + 1 / 4x + 5. This equation involves multiple equations with variables, and we need to solve for the value of x.
Step 1: Simplify the Equation
To simplify the equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 2x + 3 and 4x + 5 is (2x + 3)(4x + 5).
from sympy import symbols, Eq, solve
x = symbols('x')
eq = Eq((4x - 2) / (2x + 3), (5x + 1) / (4x + 5))
eq = Eq((4x - 2) * (4x + 5), (5x + 1) * (2x + 3))
Step 2: Expand and Simplify
After multiplying both sides of the equation by the LCM, we need to expand and simplify the equation. We can do this by multiplying out the parentheses and combining like terms.
# Expand and simplify the equation
eq = Eq(16*x**2 + 18*x - 10, 10*x**2 + 17*x + 3)
Step 3: Move All Terms to One Side
To solve for the value of x, we need to move all the terms to one side of the equation. We can do this by subtracting 10x^2 + 17x + 3 from both sides of the equation.
# Move all terms to one side
eq = Eq(6*x**2 + x - 13, 0)
Step 4: Factor the Equation
If the equation can be factored, we can use the factored form to solve for the value of x. In this case, the equation can be factored as (3x - 4)(2x + 13/3) = 0.
# Factor the equation
eq = Eq((3*x - 4)*(2*x + 13/3), 0)
Step 5: Solve for x
To solve for the value of x, we need to set each factor equal to zero and solve for x. We can do this by using the following equations:
3x - 4 = 0 --> 3x = 4 --> x = 4/3
2x + 13/3 = 0 --> 2x = -13/3 --> x = -13/6
# Solve for x
solution1 = solve(3*x - 4, x)
solution2 = solve(2*x + 13/3, x)
Conclusion
In this article, we have demonstrated the step-by-step process of solving a linear equation with multiple equations and variables. We have used the given equation 4x - 2 / 2x + 3 = 5x + 1 / 4x + 5 as an example to illustrate the process. We have shown how to simplify the equation, expand and simplify, move all terms to one side, factor the equation, and solve for the value of x. By following these steps, we can solve linear equations with multiple equations and variables.
Final Answer
The final answer is .
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Introduction
In our previous article, we demonstrated the step-by-step process of solving a linear equation with multiple equations and variables. In this article, we will provide a Q&A guide to help you better 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 simplify a linear equation?
A: To simplify a linear equation, you need to get rid of the fractions. You can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
Q: What is the least common multiple (LCM)?
A: The least common multiple (LCM) is the smallest multiple that is common to both numbers. For example, the LCM of 2 and 3 is 6.
Q: How do I expand and simplify a linear equation?
A: To expand and simplify a linear equation, you need to multiply out the parentheses and combine like terms. You can do this by using the distributive property and combining like terms.
Q: What is the distributive property?
A: The distributive property is a mathematical property that states that a(b + c) = ab + ac.
Q: How do I move all terms to one side of a linear equation?
A: To move all terms to one side of a linear equation, you need to add or subtract the same value to both sides of the equation. You can do this by using the following steps:
- Add or subtract the same value to both sides of the equation.
- Combine like terms on both sides of the equation.
- Simplify the equation.
Q: How do I factor a linear equation?
A: To factor a linear equation, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the variable. You can do this by using the following steps:
- Find two numbers whose product is equal to the constant term.
- Find two numbers whose sum is equal to the coefficient of the variable.
- Factor the equation using the two numbers.
Q: How do I solve for x in a linear equation?
A: To solve for x in a linear equation, you need to set each factor equal to zero and solve for x. You can do this by using the following steps:
- Set each factor equal to zero.
- Solve for x using the following equation: x = -b/a.
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 for x.
- Not combining like terms on both sides of the equation.
- Not factoring the equation correctly.
- Not solving for x correctly.
Conclusion
In this article, we have provided a Q&A guide to help you better understand the process of solving linear equations. We have covered topics such as simplifying linear equations, expanding and simplifying, moving all terms to one side, factoring, and solving for x. By following these steps and avoiding common mistakes, you can solve linear equations with confidence.
Final Tips
- Always simplify the equation before solving for x.
- Always combine like terms on both sides of the equation.
- Always factor the equation correctly.
- Always solve for x correctly.
By following these tips and practicing regularly, you can become proficient in solving linear equations and tackle more complex math problems with confidence.