Five Times A Number Decreased By Nine Is Equal To Twice The Number Increased By 23. Which Equation Could Be Used To Solve The Problem?A. $5x - 9 = X + 23$B. $5x - 9 = 2x + 23$C. $5x + 23 + 2x = 23$D. $5x + 23 = 2x + 23$

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Introduction

Algebraic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore how to solve a specific type of algebraic equation, focusing on the problem: "Five times a number decreased by nine is equal to twice the number increased by 23." We will examine the possible equations that can be used to solve this problem and provide a step-by-step guide on how to solve them.

Understanding the Problem

The problem states that "Five times a number decreased by nine is equal to twice the number increased by 23." Let's break this down into simpler terms:

  • "Five times a number" can be represented as 5x, where x is the unknown number.
  • "Decreased by nine" means subtracting 9 from 5x, resulting in 5x - 9.
  • "Twice the number" means multiplying the number by 2, resulting in 2x.
  • "Increased by 23" means adding 23 to 2x, resulting in 2x + 23.

So, the equation we are trying to solve is: 5x - 9 = 2x + 23.

Examining the Possible Equations

Let's examine the possible equations that can be used to solve this problem:

A. 5x−9=x+235x - 9 = x + 23

This equation is incorrect because it does not represent the problem statement. The correct equation should have 5x - 9 on the left-hand side and 2x + 23 on the right-hand side.

B. 5x−9=2x+235x - 9 = 2x + 23

This equation is correct because it represents the problem statement. We can use this equation to solve for x.

C. 5x+23+2x=235x + 23 + 2x = 23

This equation is incorrect because it does not represent the problem statement. The correct equation should have 5x - 9 on the left-hand side and 2x + 23 on the right-hand side.

D. 5x+23=2x+235x + 23 = 2x + 23

This equation is incorrect because it does not represent the problem statement. The correct equation should have 5x - 9 on the left-hand side and 2x + 23 on the right-hand side.

Solving the Equation

Now that we have identified the correct equation, let's solve for x:

5x−9=2x+235x - 9 = 2x + 23

To solve for x, we can use the following steps:

  1. Add 9 to both sides of the equation:

5x=2x+325x = 2x + 32

  1. Subtract 2x from both sides of the equation:

3x=323x = 32

  1. Divide both sides of the equation by 3:

x=32/3x = 32/3

x=10.67x = 10.67

Therefore, the value of x is 10.67.

Conclusion

Solving algebraic equations is a crucial skill for students to master. In this article, we explored how to solve a specific type of algebraic equation, focusing on the problem: "Five times a number decreased by nine is equal to twice the number increased by 23." We examined the possible equations that can be used to solve this problem and provided a step-by-step guide on how to solve them. By following these steps, students can develop their problem-solving skills and become proficient in solving algebraic equations.

Additional Resources

For students who want to practice solving algebraic equations, here are some additional resources:

  • Khan Academy: Algebra
  • Mathway: Algebra Solver
  • IXL: Algebra

By practicing and reviewing algebraic equations, students can develop their problem-solving skills and become proficient in solving algebraic equations.

Frequently Asked Questions

Q: What is an algebraic equation? A: An algebraic equation is a mathematical statement that contains variables and constants, and is used to solve for the value of the variable.

Q: How do I solve an algebraic equation? A: To solve an algebraic equation, you can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding or subtracting the same value to both sides of the equation.
  3. Divide both sides of the equation by the coefficient of the variable.

Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation is an equation that contains a single variable and has a degree of 1, while a quadratic equation is an equation that contains a single variable and has a degree of 2.

Glossary of Terms

  • Variable: A symbol or expression that represents a value that can change.
  • Constant: A value that does not change.
  • Coefficient: A number that is multiplied by a variable.
  • Degree: The highest power of the variable in an equation.

Introduction

Algebraic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will provide a comprehensive Q&A guide on algebraic equations, covering topics such as what is an algebraic equation, how to solve an algebraic equation, and more.

Q: What is an algebraic equation?

A: An algebraic equation is a mathematical statement that contains variables and constants, and is used to solve for the value of the variable.

Q: What are the different types of algebraic equations?

A: There are several types of algebraic equations, including:

  • Linear equations: Equations that contain a single variable and have a degree of 1.
  • Quadratic equations: Equations that contain a single variable and have a degree of 2.
  • Polynomial equations: Equations that contain a single variable and have a degree greater than 2.
  • Rational equations: Equations that contain a single variable and have a rational expression.

Q: How do I solve an algebraic equation?

A: To solve an algebraic equation, you can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding or subtracting the same value to both sides of the equation.
  3. Divide both sides of the equation by the coefficient of the variable.

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

A: A linear equation is an equation that contains a single variable and has a degree of 1, while a quadratic equation is an equation that contains a single variable and has a degree of 2.

Q: How do I simplify an algebraic equation?

A: To simplify an algebraic equation, you can use the following steps:

  1. Combine like terms by adding or subtracting the same value to both sides of the equation.
  2. Use the distributive property to expand expressions.
  3. Use the commutative property to rearrange terms.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single value can be multiplied by multiple values, and the result is the same as multiplying each value individually.

Q: What is the commutative property?

A: The commutative property is a mathematical property that states that the order of the terms in an expression does not change the result.

Q: How do I isolate the variable in an algebraic equation?

A: To isolate the variable in an algebraic equation, you can use the following steps:

  1. Add or subtract the same value to both sides of the equation to eliminate the constant term.
  2. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What is the coefficient of a variable?

A: The coefficient of a variable is a number that is multiplied by the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

  1. Factor the quadratic expression into two binomials.
  2. Use the quadratic formula to solve for the variable.
  3. Use the discriminant to determine the nature of the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: What is the discriminant?

A: The discriminant is a mathematical expression that is used to determine the nature of the solutions of a quadratic equation. It is given by:

b^2 - 4ac

Conclusion

Algebraic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we provided a comprehensive Q&A guide on algebraic equations, covering topics such as what is an algebraic equation, how to solve an algebraic equation, and more. By understanding these concepts and techniques, students can develop their problem-solving skills and become proficient in solving algebraic equations.

Additional Resources

For students who want to practice solving algebraic equations, here are some additional resources:

  • Khan Academy: Algebra
  • Mathway: Algebra Solver
  • IXL: Algebra

By practicing and reviewing algebraic equations, students can develop their problem-solving skills and become proficient in solving algebraic equations.

Glossary of Terms

  • Variable: A symbol or expression that represents a value that can change.
  • Constant: A value that does not change.
  • Coefficient: A number that is multiplied by a variable.
  • Degree: The highest power of the variable in an equation.
  • Linear equation: An equation that contains a single variable and has a degree of 1.
  • Quadratic equation: An equation that contains a single variable and has a degree of 2.
  • Polynomial equation: An equation that contains a single variable and has a degree greater than 2.
  • Rational equation: An equation that contains a single variable and has a rational expression.

By understanding these terms and concepts, students can develop their problem-solving skills and become proficient in solving algebraic equations.