How Many Solutions Does The Following Equation Have?$7(y-8) = 7y + 42$Choose One Answer:A. No Solutions B. Exactly One Solution C. Infinitely Many Solutions

Introduction

Linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for various fields, including algebra, geometry, and calculus. In this article, we will explore the concept of linear equations and how to determine the number of solutions for a given equation. We will use the equation $7(y-8) = 7y + 42$ as an example to illustrate the process.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation $7(y-8) = 7y + 42$

Let's examine the given equation $7(y-8) = 7y + 42$. To solve this equation, we can start by simplifying the left-hand side using the distributive property:

$$7(y-8) = 7y - 56$$

Now, we can rewrite the equation as:

$$7y - 56 = 7y + 42$$

Subtracting $7y$ from Both Sides

To isolate the variable $y$, we can subtract $7y$ from both sides of the equation:

$$-56 = 42$$

However, this is not a valid equation, as the left-hand side is a constant, and the right-hand side is also a constant. This means that the equation has no solution.

Understanding the Number of Solutions

In general, a linear equation can have one of three possible solutions:

• No solutions: This occurs when the equation is inconsistent, meaning that it is impossible to find a value for the variable that satisfies the equation.
• Exactly one solution: This occurs when the equation is consistent, meaning that there is a unique value for the variable that satisfies the equation.
• Infinitely many solutions: This occurs when the equation is an identity, meaning that it is true for all values of the variable.

Conclusion

In conclusion, the equation $7(y-8) = 7y + 42$ has no solution. This is because the equation is inconsistent, and it is impossible to find a value for the variable $y$ that satisfies the equation. Understanding the number of solutions for a linear equation is crucial for solving problems in mathematics and other fields.

Key Takeaways

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• The equation $7(y-8) = 7y + 42$ has no solution.
• A linear equation can have one of three possible solutions: no solutions, exactly one solution, or infinitely many solutions.

Further Reading

For more information on linear equations and how to solve them, we recommend the following resources:

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• Wolfram Alpha: Linear Equations

Introduction

Linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for various fields, including algebra, geometry, and calculus. In our previous article, we explored the concept of linear equations and how to determine the number of solutions for a given equation. In this article, we will answer some frequently asked questions about linear equations and provide additional guidance on how to solve them.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that 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 can use various methods, including algebraic manipulation, graphing, and substitution. Here are the steps to follow:

1. Simplify the equation: Start by simplifying the equation by combining like terms and eliminating any parentheses.
2. Isolate the variable: Use algebraic manipulation to isolate the variable on one side of the equation.
3. Check the solution: Once you have isolated the variable, check your solution by plugging it 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. For example, the equation $x + 2 = 5$ is a linear equation, while the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: Can a linear equation have more than one solution?

A: No, a linear equation can have at most one solution. If a linear equation has no solution, it means that the equation is inconsistent, and there is no value for the variable that satisfies the equation. If a linear equation has exactly one solution, it means that the equation is consistent, and there is a unique value for the variable that satisfies the equation.

Q: How do I determine the number of solutions for a linear equation?

A: To determine the number of solutions for a linear equation, you can use the following steps:

1. Check if the equation is consistent: Check if the equation is consistent by plugging in a value for the variable and checking if the equation is true.
2. Check if the equation is inconsistent: Check if the equation is inconsistent by plugging in a value for the variable and checking if the equation is false.
3. Determine the number of solutions: If the equation is consistent, it means that there is a unique value for the variable that satisfies the equation, and the equation has exactly one solution. If the equation is inconsistent, it means that there is no value for the variable that satisfies the equation, and the equation has no solution.

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

A: Here are some common mistakes to avoid when solving linear equations:

• Not simplifying the equation: Failing to simplify the equation can make it difficult to isolate the variable and find the solution.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.
• Not using the correct method: Using the wrong method to solve the equation can lead to incorrect answers.

Conclusion

In conclusion, solving linear equations is a crucial skill in mathematics and other fields. By understanding the concept of linear equations and how to determine the number of solutions, you can solve problems with confidence. Remember to simplify the equation, isolate the variable, and check the solution to ensure that you are getting the correct answer.

Key Takeaways

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• To solve a linear equation, simplify the equation, isolate the variable, and check the solution.
• A linear equation can have at most one solution.
• To determine the number of solutions for a linear equation, check if the equation is consistent or inconsistent.

Further Reading

For more information on linear equations and how to solve them, we recommend the following resources:

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• Wolfram Alpha: Linear Equations

By following these steps and avoiding common mistakes, you can become proficient in solving linear equations and tackle more complex problems with confidence.