In The Equation Below, Solve For The Value Of $y$:$\frac{3}{2} Y + 5 = Y$

In the equation $\frac{3}{2} y + 5 = y$, we are tasked with solving for the value of $y$. This equation is a linear equation, which means it can be solved using basic algebraic techniques.

Understanding the Equation

The given equation is $\frac{3}{2} y + 5 = y$. To solve for $y$, we need to isolate the variable $y$ on one side of the equation. The equation consists of two terms: $\frac{3}{2} y$ and $y$. The goal is to get rid of the fraction and combine like terms.

Step 1: Subtract $\frac{3}{2} y$ from both sides

To eliminate the fraction, we can multiply both sides of the equation by the denominator of the fraction, which is $2$. This will get rid of the fraction and allow us to work with whole numbers.

\frac{3}{2} y + 5 = y
\implies 2 \times \left(\frac{3}{2} y + 5\right) = 2 \times y
\implies 3y + 10 = 2y

Step 2: Subtract $2y$ from both sides

Now that we have eliminated the fraction, we can focus on isolating the variable $y$. We can do this by subtracting $2y$ from both sides of the equation.

3y + 10 = 2y
\implies 3y - 2y + 10 = 2y - 2y
\implies y + 10 = 0

Step 3: Subtract 10 from both sides

We are getting close to solving for $y$. To isolate the variable, we can subtract 10 from both sides of the equation.

y + 10 = 0
\implies y + 10 - 10 = 0 - 10
\implies y = -10

Conclusion

In the equation $\frac{3}{2} y + 5 = y$, we have successfully solved for the value of $y$. By following the steps outlined above, we have isolated the variable $y$ and found that $y = -10$.

Real-World Applications

Solving linear equations like the one above has numerous real-world applications. For example, in finance, linear equations can be used to calculate interest rates, investment returns, and other financial metrics. In physics, linear equations can be used to model the motion of objects, calculate forces, and determine energy levels.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to isolate the variable on one side of the equation. Additionally, be careful when multiplying or dividing both sides of the equation by a fraction, as this can lead to incorrect solutions.

Common Mistakes

When solving linear equations, some common mistakes to avoid include:

• Not following the order of operations (PEMDAS)
• Not isolating the variable on one side of the equation
• Multiplying or dividing both sides of the equation by a fraction without considering the implications
• Not checking the solution for validity

Conclusion

In the previous article, we discussed how to solve linear equations like $\frac{3}{2} y + 5 = y$. However, we understand that you may still have some questions about solving linear equations. In this article, we will address some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

A linear equation is an equation in which the highest power of the variable (usually $x$ or $y$) is 1. In other words, a linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I know if an equation is linear?

To determine if an equation is linear, look for the following characteristics:

• The highest power of the variable is 1.
• The equation can be written in the form $ax + b = c$.
• The equation does not contain any fractions or decimals.

Q: What are some common types of linear equations?

Some common types of linear equations include:

• Simple linear equations: Equations of the form $ax + b = c$, where $a$, $b$, and $c$ are constants.
• Linear equations with fractions: Equations of the form $\frac{a}{b} x + c = d$, where $a$, $b$, $c$, and $d$ are constants.
• Linear equations with decimals: Equations of the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $a$ or $b$ is a decimal.

Q: How do I solve a linear equation with fractions?

To solve a linear equation with fractions, follow these steps:

1. Multiply both sides of the equation by the denominator of the fraction.
2. Simplify the equation by combining like terms.
3. Isolate the variable on one side of the equation.

Q: How do I solve a linear equation with decimals?

To solve a linear equation with decimals, follow these steps:

1. Multiply both sides of the equation by the decimal to eliminate it.
2. Simplify the equation by combining like terms.
3. Isolate the variable on one side of the equation.

Q: What are some common mistakes to avoid when solving linear equations?

Some common mistakes to avoid when solving linear equations include:

• Not following the order of operations (PEMDAS).
• Not isolating the variable on one side of the equation.
• Multiplying or dividing both sides of the equation by a fraction without considering the implications.
• Not checking the solution for validity.

Q: How do I check the solution for validity?

To check the solution for validity, plug the solution back into the original equation and verify that it is true. If the solution is not valid, re-evaluate the equation and try again.

Q: What are some real-world applications of linear equations?

Some real-world applications of linear equations include:

• Finance: Linear equations can be used to calculate interest rates, investment returns, and other financial metrics.
• Physics: Linear equations can be used to model the motion of objects, calculate forces, and determine energy levels.
• Engineering: Linear equations can be used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, solving linear equations requires careful attention to detail and a solid understanding of algebraic techniques. By following the steps outlined above and avoiding common mistakes, you can successfully solve linear equations and apply them to real-world problems.