Solve The Equation: + 2 4 − Y − 6 3 = 1 2 \frac{+2}{4} - \frac{y-6}{3} = \frac{1}{2} 4 + 2 − 3 Y − 6 = 2 1

Mar 13, 2025 by ADMIN 113 views

$Solving the Equation: $\frac{+2}{4} - \frac{y-6}{3} = \frac{1}{2}$$

Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach and solve various types of equations. In this article, we will focus on solving a specific equation involving fractions. The equation we will be solving is $\frac{+2}{4} - \frac{y-6}{3} = \frac{1}{2}$ . This equation involves fractions, and we will use algebraic techniques to solve for the variable $y$ .

Understanding the Equation

Before we start solving the equation, let's break it down and understand what it means. The equation is a linear equation, which means it can be written in the form $ax + b = c$ , where $a$ , $b$ , and $c$ are constants. In this case, the equation is $\frac{+2}{4} - \frac{y-6}{3} = \frac{1}{2}$ . We can see that the equation involves fractions, and we need to find the value of $y$ that satisfies the equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by getting 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 4 and 3 is 12. So, we can multiply both sides of the equation by 12 to get rid of the fractions.

from sympy import symbols, Eq, solve
y = symbols('y')

equation = Eq((2/4) - (y-6)/3, 1/2)

equation = Eq(12 * ((2/4) - (y-6)/3), 12 * (1/2))

Step 2: Distribute the Multiplication

Now that we have multiplied both sides of the equation by 12, we can distribute the multiplication to simplify the equation. We can do this by multiplying each term inside the parentheses by 12.

# Distribute the multiplication
equation = Eq(12 * (2/4) - 12 * (y-6)/3, 12 * (1/2))

Step 3: Simplify the Fractions

Now that we have distributed the multiplication, we can simplify the fractions. We can do this by dividing each term by the denominator.

# Simplify the fractions
equation = Eq(3 - 4 * (y-6)/3, 6)

Step 4: Multiply Both Sides by 3

To get rid of the fraction, we can multiply both sides of the equation by 3.

# Multiply both sides by 3
equation = Eq(3 * (3 - 4 * (y-6)/3), 3 * 6)

Step 5: Simplify the Equation

Now that we have multiplied both sides of the equation by 3, we can simplify the equation. We can do this by multiplying each term inside the parentheses by 3.

# Simplify the equation
equation = Eq(9 - 4 * (y-6), 18)

Step 6: Distribute the Multiplication

Now that we have simplified the equation, we can distribute the multiplication to simplify further.

# Distribute the multiplication
equation = Eq(9 - 4 * y + 24, 18)

Step 7: Combine Like Terms

Now that we have distributed the multiplication, we can combine like terms to simplify the equation further.

# Combine like terms
equation = Eq(33 - 4 * y, 18)

Step 8: Subtract 33 from Both Sides

To isolate the term with the variable, we can subtract 33 from both sides of the equation.

# Subtract 33 from both sides
equation = Eq(-4 * y, -15)

Step 9: Divide Both Sides by -4

Finally, we can divide both sides of the equation by -4 to solve for the variable $y$ .

# Divide both sides by -4
equation = Eq(y, 15/4)

Conclusion

In this article, we solved the equation $\frac{+2}{4} - \frac{y-6}{3} = \frac{1}{2}$ using algebraic techniques. We started by simplifying the equation, distributing the multiplication, and combining like terms. Finally, we isolated the term with the variable and solved for $y$ . The solution to the equation is $y = \frac{15}{4}$ .

Final Answer

The final answer is $\boxed{\frac{15}{4}}$ .

Introduction

In our previous article, we solved the equation $\frac{+2}{4} - \frac{y-6}{3} = \frac{1}{2}$ using algebraic techniques. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify the equation by getting rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, we need to list the multiples of each denominator and find the smallest multiple that is common to both. In this case, the LCM of 4 and 3 is 12.

Q: What is the next step after simplifying the equation?

A: After simplifying the equation, we need to distribute the multiplication to simplify further. We can do this by multiplying each term inside the parentheses by the number outside the parentheses.

Q: How do I distribute the multiplication?

A: To distribute the multiplication, we need to multiply each term inside the parentheses by the number outside the parentheses. For example, if we have the equation $2(x+3)$ , we would multiply each term inside the parentheses by 2 to get $2x+6$ .

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to isolate the term with the variable and solve for $y$ . We can do this by subtracting 33 from both sides of the equation and then dividing both sides by -4.

Q: What is the solution to the equation?

A: The solution to the equation is $y = \frac{15}{4}$ .

Q: Can I use a calculator to solve the equation?

A: Yes, you can use a calculator to solve the equation. However, it's always a good idea to check your work by plugging the solution back into the original equation.

Q: What if I get stuck while solving the equation?

A: If you get stuck while solving the equation, don't worry! You can try breaking down the problem into smaller steps or seeking help from a teacher or tutor.

Conclusion

In this Q&A article, we provided answers to common questions that readers may have while solving the equation $\frac{+2}{4} - \frac{y-6}{3} = \frac{1}{2}$ . We hope that this article has helped to clarify any doubts or questions that readers may have.

Final Answer

The final answer is $\boxed{\frac{15}{4}}$ .