Solve This Equation: 80 = 3 Y + 2 Y + 4 + 1 80 = 3y + 2y + 4 + 1 80 = 3 Y + 2 Y + 4 + 1 .A. Y = 15 Y = 15 Y = 15 B. Y = 1 5 Y = \frac{1}{5} Y = 5 1 ​ C. Y = 75 Y = 75 Y = 75 D. Y = − 15 Y = -15 Y = − 15

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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 linear equation, 80=3y+2y+4+180 = 3y + 2y + 4 + 1, and explore the different methods and techniques used to find the solution.

Understanding the Equation

The given equation is a linear equation in one variable, yy. It can be written in the standard form as:

80=3y+2y+4+180 = 3y + 2y + 4 + 1

To solve this equation, we need to isolate the variable yy on one side of the equation.

Combining Like Terms

The first step in solving the equation is to combine like terms. In this case, we have two terms with the variable yy, which are 3y3y and 2y2y. We can combine these terms by adding their coefficients:

3y+2y=5y3y + 2y = 5y

Now, the equation becomes:

80=5y+4+180 = 5y + 4 + 1

Simplifying the Equation

The next step is to simplify the equation by combining the constant terms on the right-hand side:

4+1=54 + 1 = 5

So, the equation becomes:

80=5y+580 = 5y + 5

Isolating the Variable

To isolate the variable yy, we need to get rid of the constant term on the right-hand side. We can do this by subtracting 5 from both sides of the equation:

80−5=5y+5−580 - 5 = 5y + 5 - 5

This simplifies to:

75=5y75 = 5y

Solving for yy

Now that we have isolated the variable yy, we can solve for its value. To do this, we need to divide both sides of the equation by 5:

755=5y5\frac{75}{5} = \frac{5y}{5}

This simplifies to:

15=y15 = y

Conclusion

In this article, we solved the linear equation 80=3y+2y+4+180 = 3y + 2y + 4 + 1 using the method of combining like terms, simplifying the equation, and isolating the variable. We found that the value of yy is 15.

Answer Key

The correct answer is:

A. y=15y = 15

Tips and Tricks

  • When solving linear equations, it's essential to combine like terms and simplify the equation to make it easier to solve.
  • Isolating the variable is a crucial step in solving linear equations. Make sure to get rid of any constant terms on the right-hand side.
  • When dividing both sides of the equation by a coefficient, make sure to divide both sides by the same value.

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. Some examples include:

  • Modeling population growth and decline
  • Calculating the cost of goods and services
  • Determining the trajectory of objects in motion
  • Solving optimization problems

Practice Problems

Try solving the following linear equations:

  1. 2x+5=112x + 5 = 11
  2. x−3=7x - 3 = 7
  3. 4y+2=144y + 2 = 14

Conclusion

Solving linear equations is a fundamental skill that has numerous applications in various fields. By following the steps outlined in this article, you can solve linear equations with ease and confidence. Remember to combine like terms, simplify the equation, and isolate the variable to find the solution.

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Introduction

In our previous article, we explored the concept of solving linear equations and provided a step-by-step guide on how to solve a specific equation. In this article, we will address some of the most frequently asked questions about solving linear equations.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It can be written in the standard form as:

ax+b=cax + b = c

where aa, bb, and cc are constants, and xx is the variable.

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

A: 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 standard form.
  • The equation has a constant term on the right-hand side.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation has a highest power of 1, while a quadratic equation has a highest power of 2. For example:

  • Linear equation: 2x+3=52x + 3 = 5
  • Quadratic equation: x2+4x+4=0x^2 + 4x + 4 = 0

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

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

  1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
  2. Simplify the equation.
  3. Isolate the variable.

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

A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the concept and steps involved in solving linear equations, as calculators can only provide the solution, not the process.

Q: What is the order of operations when solving linear equations?

A: When solving linear equations, follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: Can I solve linear equations with variables on both sides?

A: Yes, you can solve linear equations with variables on both sides. To do this, follow these steps:

  1. Add or subtract the same value to both sides of the equation to eliminate the variable on one side.
  2. Isolate the variable on one side of the equation.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more linear equations with the same variables.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS).
  • Use a calculator to check your solution, but understand the concept and steps involved in solving linear equations.
  • Practice solving linear equations with different types of coefficients and variables.

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. Some examples include:

  • Modeling population growth and decline
  • Calculating the cost of goods and services
  • Determining the trajectory of objects in motion
  • Solving optimization problems

Practice Problems

Try solving the following linear equations:

  1. 2x3+2=5\frac{2x}{3} + 2 = 5
  2. x−2=3x - 2 = 3
  3. 4y+2=144y + 2 = 14

Conclusion

Solving linear equations is a fundamental skill that has numerous applications in various fields. By understanding the concept and steps involved in solving linear equations, you can tackle a wide range of problems and challenges. Remember to follow the order of operations (PEMDAS), use a calculator to check your solution, and practice solving linear equations with different types of coefficients and variables.