Solve For $x$ To Determine If The Equation Has One, None, Or Infinite Solutions.$5(2x + 2) - 7x = 3x + 6$A. The Equation Has One Solution: $x = 10$.B. The Equation Has No Solution.C. The Equation Has Infinite Solutions.D. The

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, 5(2x+2)−7x=3x+65(2x + 2) - 7x = 3x + 6, to determine if it has one, none, or infinite solutions. We will break down the solution process into manageable steps, making it easy to understand and follow.

Understanding the Equation

The given equation is 5(2x+2)−7x=3x+65(2x + 2) - 7x = 3x + 6. To solve this equation, we need to simplify it by applying the distributive property and combining like terms.

Distributive Property

The distributive property states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b + c) = ab + ac. We can apply this property to the given equation:

5(2x+2)−7x=3x+65(2x + 2) - 7x = 3x + 6

Using the distributive property, we can rewrite the equation as:

10x+10−7x=3x+610x + 10 - 7x = 3x + 6

Combining Like Terms

Now that we have applied the distributive property, we can combine like terms to simplify the equation:

10x+10−7x=3x+610x + 10 - 7x = 3x + 6

Combining like terms, we get:

3x+10=3x+63x + 10 = 3x + 6

Solving for xx

Now that we have simplified the equation, we can solve for xx. To do this, we need to isolate the variable xx on one side of the equation.

Subtracting 3x3x from Both Sides

We can start by subtracting 3x3x from both sides of the equation:

3x+10−3x=3x+6−3x3x + 10 - 3x = 3x + 6 - 3x

This simplifies to:

10=610 = 6

Conclusion

Unfortunately, we have reached a dead end. The equation 10=610 = 6 is a contradiction, as 1010 is not equal to 66. This means that the original equation 5(2x+2)−7x=3x+65(2x + 2) - 7x = 3x + 6 has no solution.

Conclusion

In conclusion, we have solved the linear equation 5(2x+2)−7x=3x+65(2x + 2) - 7x = 3x + 6 and determined that it has no solution. This is because the equation leads to a contradiction, 10=610 = 6, which is not possible.

Frequently Asked Questions

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 is 1, whereas a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I know if an equation has one, none, or infinite solutions?

A: To determine the number of solutions, you need to solve the equation and check if it leads to a contradiction, a single value, or multiple values.

Q: Can I use the same method to solve quadratic equations?

A: No, the method used to solve linear equations is not applicable to quadratic equations. Quadratic equations require a different approach, such as factoring or using the quadratic formula.

Final Thoughts

Solving linear equations is an essential skill for students to master. By following the steps outlined in this article, you can determine if an equation has one, none, or infinite solutions. Remember to apply the distributive property and combine like terms to simplify the equation, and then solve for the variable. With practice and patience, you will become proficient in solving linear equations and be able to tackle more complex equations in the future.

References

Additional Resources

Discussion

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Share Your Thoughts

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  • Have you encountered any challenges while solving linear equations?
  • Do you have any tips or tricks for solving linear equations?

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Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving them. In our previous article, we provided a step-by-step guide on solving linear equations. However, we know that sometimes, it's not enough to just provide a guide; you need to have a deeper understanding of the concepts involved. That's why we've created this Q&A article, where we'll answer 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 form of ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I know if an equation is linear or not?

A: To determine if an equation is linear or not, you need to check if the highest power of the variable is 1. If it is, then the equation is linear. If not, then it's not linear.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c, a(b + c) = ab + ac. It's used to simplify equations by distributing the coefficients to the terms inside the parentheses.

Q: How do I apply the distributive property to an equation?

A: To apply the distributive property to an equation, you need to multiply the coefficient outside the parentheses to each term inside the parentheses. For example, if you have the equation 2(x + 3), you would multiply 2 to each term inside the parentheses, resulting in 2x + 6.

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 is 1, whereas a quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation x + 2 = 3 is a linear equation, while the equation x^2 + 2x + 1 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is as follows:

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

Q: How do I use the order of operations to solve an equation?

A: To use the order of operations to solve an equation, you need to follow the order of operations and perform the operations in the correct order. For example, if you have the equation 2(x + 3) - 4, you would first evaluate the expression inside the parentheses, then multiply 2 to the result, and finally subtract 4.

Q: What is a contradiction?

A: A contradiction is a statement that is false. In the context of solving linear equations, a contradiction occurs when the equation leads to a statement that is false, such as 2 = 3.

Q: What does it mean when an equation has no solution?

A: When an equation has no solution, it means that the equation is a contradiction, and there is no value of the variable that can make the equation true.

Q: What does it mean when an equation has infinite solutions?

A: When an equation has infinite solutions, it means that the equation is an identity, and any value of the variable will make the equation true.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving them. By following the order of operations and applying the distributive property, you can simplify equations and solve for the variable. Remember, a linear equation is an equation in which the highest power of the variable is 1, and it can be written in the form of ax + b = c. With practice and patience, you'll become proficient in solving linear equations and be able to tackle more complex equations in the future.

Frequently Asked Questions

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 is 1, whereas a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I know if an equation is linear or not?

A: To determine if an equation is linear or not, you need to check if the highest power of the variable is 1. If it is, then the equation is linear. If not, then it's not linear.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c, a(b + c) = ab + ac. It's used to simplify equations by distributing the coefficients to the terms inside the parentheses.

Final Thoughts

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving them. By following the order of operations and applying the distributive property, you can simplify equations and solve for the variable. Remember, a linear equation is an equation in which the highest power of the variable is 1, and it can be written in the form of ax + b = c. With practice and patience, you'll become proficient in solving linear equations and be able to tackle more complex equations in the future.

References

Additional Resources

Discussion

Do you have any questions or comments about solving linear equations? Share your thoughts and experiences in the discussion section below.

Share Your Thoughts

  • Do you have any questions about solving linear equations?
  • Have you encountered any challenges while solving linear equations?
  • Do you have any tips or tricks for solving linear equations?

Comment Below

We welcome your comments and feedback. Please share your thoughts and experiences in the discussion section below.