Solve For $y$.$-\frac{1}{2} - \frac{4}{5} Y = \frac{2}{3}$Simplify Your Answer As Much As Possible.

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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 fractions. We will use the given equation $-\frac{1}{2} - \frac{4}{5} y = \frac{2}{3}$ as an example and walk through the steps to simplify and solve for $y$.

Understanding the Equation

Before we dive into the solution, let's take a closer look at the equation. We have a linear equation with three terms: $-\frac{1}{2}$, $-\frac{4}{5} y$, and $\frac{2}{3}$. The goal is to isolate the variable $y$ and simplify the equation as much as possible.

Step 1: Eliminate the Fractions

To eliminate the fractions, we need to find the least common multiple (LCM) of the denominators. In this case, the denominators are $2$, $5$, and $3$. The LCM of these numbers is $30$. We can multiply both sides of the equation by $30$ to eliminate the fractions.

30×(−12−45y=23)30 \times \left(-\frac{1}{2} - \frac{4}{5} y = \frac{2}{3}\right)

⇒−15−24y=20\Rightarrow -15 - 24y = 20

Step 2: Isolate the Variable

Now that we have eliminated the fractions, we can focus on isolating the variable $y$. We want to get $y$ by itself on one side of the equation. To do this, we can add $24y$ to both sides of the equation and then subtract $20$ from both sides.

−15−24y+24y=20+24y−24y-15 - 24y + 24y = 20 + 24y - 24y

⇒−15=20\Rightarrow -15 = 20

Step 3: Simplify the Equation

At this point, we have a simple equation: $-15 = 20$. However, this equation is not true, which means that the original equation $-\frac{1}{2} - \frac{4}{5} y = \frac{2}{3}$ has no solution. This is because the left-hand side of the equation is always negative, while the right-hand side is always positive.

Conclusion

In this article, we walked through the steps to solve a linear equation with fractions. We eliminated the fractions by multiplying both sides of the equation by the least common multiple of the denominators and then isolated the variable $y$ by adding and subtracting terms. However, we found that the original equation has no solution, which means that the equation is inconsistent.

Tips and Tricks

  • When solving linear equations with fractions, it's essential to eliminate the fractions first by finding the least common multiple of the denominators.
  • To isolate the variable, add or subtract terms to get the variable by itself on one side of the equation.
  • If the equation has no solution, it means that the equation is inconsistent, and there is no value of the variable that satisfies the equation.

Common Mistakes

  • Failing to eliminate the fractions before isolating the variable.
  • Not checking if the equation has a solution before solving it.
  • Not simplifying the equation after isolating the variable.

Real-World Applications

Linear equations with fractions are used in various real-world applications, such as:

  • Finance: Calculating interest rates and investment returns.
  • Science: Modeling population growth and chemical reactions.
  • Engineering: Designing electrical circuits and mechanical systems.

Final Thoughts

Introduction

In our previous article, we walked through the steps to solve a linear equation with fractions. However, we received many questions from readers who wanted more clarification and examples. In this article, we will address some of the most frequently asked questions about solving linear equations with fractions.

Q: What is the least common multiple (LCM) of the denominators?

A: The least common multiple (LCM) of the denominators is the smallest number that is a multiple of all the denominators. For example, if we have the denominators 2, 3, and 4, the LCM is 12.

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

A: To find the LCM of the denominators, you can list the multiples of each denominator and find the smallest number that appears in all the lists. Alternatively, you can use the following formula:

LCM(a, b, c) = (a × b × c) / (gcd(a, b) × gcd(b, c) × gcd(c, a))

where gcd(a, b) is the greatest common divisor of a and b.

Q: What if the LCM is not a whole number?

A: If the LCM is not a whole number, you can multiply both sides of the equation by the LCM to eliminate the fractions. For example, if the LCM is 3.5, you can multiply both sides of the equation by 3.5 to get rid of the fractions.

Q: Can I use a calculator to find the LCM?

A: Yes, you can use a calculator to find the LCM. Most calculators have a built-in function to find the LCM of two or more numbers.

Q: How do I isolate the variable in a linear equation with fractions?

A: To isolate the variable in a linear equation with fractions, you can add or subtract terms to get the variable by itself on one side of the equation. For example, if we have the equation 2x + 3 = 5, we can subtract 3 from both sides to get 2x = 2.

Q: What if the equation has no solution?

A: If the equation has no solution, it means that the equation is inconsistent, and there is no value of the variable that satisfies the equation. This can happen if the left-hand side and right-hand side of the equation are not equal.

Q: Can I use a graphing calculator to solve linear equations with fractions?

A: Yes, you can use a graphing calculator to solve linear equations with fractions. Graphing calculators can help you visualize the equation and find the solution.

Q: Are there any shortcuts or tricks to solving linear equations with fractions?

A: Yes, there are several shortcuts and tricks to solving linear equations with fractions. One common trick is to multiply both sides of the equation by the least common multiple of the denominators. Another trick is to use the distributive property to simplify the equation.

Q: Can I use linear equations with fractions to model real-world problems?

A: Yes, linear equations with fractions can be used to model real-world problems. For example, you can use linear equations with fractions to model population growth, chemical reactions, and electrical circuits.

Conclusion

Solving linear equations with fractions requires careful attention to detail and a step-by-step approach. By following the steps outlined in this article, you can simplify and solve these types of equations. Remember to eliminate the fractions first, isolate the variable, and check if the equation has a solution. With practice and patience, you'll become proficient in solving linear equations with fractions.

Tips and Tricks

  • Use a calculator to find the LCM of the denominators.
  • Multiply both sides of the equation by the LCM to eliminate the fractions.
  • Use the distributive property to simplify the equation.
  • Graph the equation on a graphing calculator to visualize the solution.

Common Mistakes

  • Failing to eliminate the fractions before isolating the variable.
  • Not checking if the equation has a solution before solving it.
  • Not simplifying the equation after isolating the variable.

Real-World Applications

Linear equations with fractions are used in various real-world applications, such as:

  • Finance: Calculating interest rates and investment returns.
  • Science: Modeling population growth and chemical reactions.
  • Engineering: Designing electrical circuits and mechanical systems.

Final Thoughts

Solving linear equations with fractions requires practice and patience. By following the steps outlined in this article and using the tips and tricks provided, you can become proficient in solving these types of equations. Remember to eliminate the fractions first, isolate the variable, and check if the equation has a solution. With time and practice, you'll become a master of solving linear equations with fractions.