Find The Value Of The Unknown Variable In This Equation:a) 2 Z − 1 3 − Z − 7 5 = 2 \frac{2z-1}{3}-\frac{z-7}{5}=2 3 2 Z − 1 − 5 Z − 7 = 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 type of linear equation, which involves finding the value of an unknown variable. We will use a step-by-step approach to solve the equation, and provide explanations and examples to help readers understand the concept.
The Equation
The equation we will be solving is:
This equation involves two fractions, and our goal is to find the value of the unknown variable .
Step 1: Multiply Both Sides by the Least Common Multiple (LCM)
To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 3 and 5 is 15.
# Multiply both sides by 15
15 * ((2z-1)/3 - (z-7)/5) = 15 * 2
Step 2: Distribute the Multiplication
Now, we need to distribute the multiplication to both terms on the left-hand side of the equation.
# Distribute the multiplication
(15 * (2z-1))/3 - (15 * (z-7))/5 = 30
Step 3: Simplify the Fractions
We can simplify the fractions by dividing the numerator and denominator by their greatest common divisor (GCD).
# Simplify the fractions
5(2z-1) - 3(z-7) = 30
Step 4: Expand and Simplify
Now, we need to expand and simplify the equation by multiplying the terms inside the parentheses.
# Expand and simplify
10z - 5 - 3z + 21 = 30
Step 5: Combine Like Terms
We can combine like terms by adding or subtracting the coefficients of the same variable.
# Combine like terms
7z + 16 = 30
Step 6: Isolate the Variable
Now, we need to isolate the variable by subtracting 16 from both sides of the equation.
# Isolate the variable
7z = 14
Step 7: Solve for
Finally, we can solve for by dividing both sides of the equation by 7.
# Solve for z
z = 14/7
Conclusion
In this article, we solved a linear equation involving two fractions. We used a step-by-step approach to eliminate the fractions, combine like terms, and isolate the variable. The final solution is . This equation is a simple example of a linear equation, and solving it requires a basic understanding of algebraic concepts.
Real-World Applications
Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, linear equations can be used to model the motion of objects, while in engineering, they can be used to design and optimize systems. In economics, linear equations can be used to model the behavior of markets and economies, and in computer science, they can be used to develop algorithms and solve problems.
Tips and Tricks
Here are some tips and tricks to help you solve linear equations:
- Use the distributive property: When multiplying a term by a binomial, use the distributive property to expand the product.
- Combine like terms: When adding or subtracting terms, combine like terms to simplify the equation.
- Isolate the variable: When solving for a variable, isolate it by subtracting or adding terms to both sides of the equation.
- Check your work: When solving a linear equation, check your work by plugging the solution back into the original equation.
Common Mistakes
Here are some common mistakes to avoid when solving linear equations:
- Forgetting to distribute: When multiplying a term by a binomial, don't forget to distribute the multiplication.
- Not combining like terms: When adding or subtracting terms, don't forget to combine like terms.
- Not isolating the variable: When solving for a variable, don't forget to isolate it by subtracting or adding terms to both sides of the equation.
- Not checking your work: When solving a linear equation, don't forget to check your work by plugging the solution back into the original equation.
Conclusion
Solving linear equations is a crucial skill for students and professionals alike. By following a step-by-step approach and using algebraic concepts, we can solve linear equations involving fractions, decimals, and variables. Remember to use the distributive property, combine like terms, isolate the variable, and check your work to ensure accuracy. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. It 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 isolate the variable by performing operations on both sides of the equation. This can involve adding, subtracting, multiplying, or dividing both sides of the equation.
Q: What is the order of operations when solving a linear equation?
A: The order of operations when solving a linear equation is:
- Parentheses: Evaluate 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 in a linear equation?
A: When a linear equation involves fractions, you can eliminate the fractions 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) of two or more numbers is the smallest number that is a multiple of each of the given numbers.
Q: How do I find the LCM of two numbers?
A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that appears in both lists.
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 work by plugging the solution back into the original 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.
Q: How do I solve a quadratic equation?
A: To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that can be used to solve quadratic equations. It is: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.
Q: Can I use a graphing calculator to solve a linear equation?
A: Yes, you can use a graphing calculator to solve a linear equation. However, it's always a good idea to check your work by plugging the solution back into the original equation.
Q: What is the difference between a linear equation and a system of linear equations?
A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a system of linear equations is a set of two or more linear equations that are solved simultaneously.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you can use the method of substitution or the method of elimination.
Q: What is the method of substitution?
A: The method of substitution is a technique used to solve a system of linear equations by substituting one equation into the other equation.
Q: What is the method of elimination?
A: The method of elimination is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables.
Q: Can I use a computer program to solve a system of linear equations?
A: Yes, you can use a computer program to solve a system of linear equations. However, it's always a good idea to check your work by plugging the solution back into the original equations.
Conclusion
Solving linear equations is a crucial skill for students and professionals alike. By following a step-by-step approach and using algebraic concepts, we can solve linear equations involving fractions, decimals, and variables. Remember to use the distributive property, combine like terms, isolate the variable, and check your work to ensure accuracy. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.