Select The Correct Answer.Which Statement Is True About This Equation?$\[-9(x+3)+12=-3(2x+5)-3x\\]A. The Equation Has One Solution, $\[x=1\\].B. The Equation Has One Solution, $\[x=0\\].C. The Equation Has No Solution.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 explore how to solve linear equations, with a focus on the given equation: ${-9(x+3)+12=-3(2x+5)-3x}$. We will examine each statement and determine which one is true about this equation.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand its components. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, xx) is 1. The equation is also a multi-step equation, which means it requires multiple steps to solve.

Step 1: Distribute the Negative Sign

The first step in solving the equation is to distribute the negative sign to the terms inside the parentheses. This will give us:

−9(x+3)+12=−3(2x+5)−3x-9(x+3)+12=-3(2x+5)-3x

−9x−27+12=−6x−15−3x-9x-27+12=-6x-15-3x

Step 2: Combine Like Terms

The next step is to combine like terms. We can combine the constants and the xx terms separately:

−9x−15=−6x−3x-9x-15=-6x-3x

−9x−15=−9x-9x-15=-9x

Step 3: Simplify the Equation

Now that we have combined like terms, we can simplify the equation by canceling out the −9x-9x terms:

−15=−15-15=-15

Conclusion

As we can see, the equation simplifies to −15=−15-15=-15, which is a true statement. This means that the equation has an infinite number of solutions, not just one solution. Therefore, statement A and B are incorrect, and statement C is also incorrect because the equation does have a solution.

The Correct Answer

The correct answer is D. The equation has an infinite number of solutions.

Why is this the case?

The reason why the equation has an infinite number of solutions is that the equation is an identity. An identity is an equation that is true for all values of the variable. In this case, the equation is true for all values of xx, which means that there are an infinite number of solutions.

Real-World Applications

Solving linear equations is a crucial skill in many real-world applications, such as:

  • Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of objects.
  • Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
  • Economics: Linear equations are used to model economic systems, including the supply and demand of goods and services.

Conclusion

In conclusion, solving linear equations is a crucial skill that has many real-world applications. By following the steps outlined in this article, we can solve linear equations and determine which statement is true about the given equation. The correct answer is D. The equation has an infinite number of solutions.

Final Thoughts

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, xx) is 1. It is a fundamental concept in mathematics and is used to describe many real-world applications.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow these steps:

  1. Distribute the negative sign: If the equation has a negative sign in front of the parentheses, distribute it to the terms inside the parentheses.
  2. Combine like terms: Combine the constants and the xx terms separately.
  3. Simplify the equation: Cancel out any terms that are the same on both sides of the equation.
  4. Check your solution: Plug your solution back into the original equation to make sure it is true.

Q: What is an identity?

A: An identity is an equation that is true for all values of the variable. In other words, it is an equation that is always true, regardless of the value of the variable.

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

A: To determine if an equation is an identity, try plugging in different values for the variable and see if the equation is always true. If it is, then the equation is an identity.

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, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, x+3=5x+3=5 is a linear equation, while x2+4x+4=0x^2+4x+4=0 is a quadratic equation.

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

A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your solution by plugging it back into the original equation to make sure it is true.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of objects.
  • Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
  • Economics: Linear equations are used to model economic systems, including the supply and demand of goods and services.

Q: Can I use linear equations to solve problems in other areas of mathematics?

A: Yes, linear equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

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

A: Some common mistakes to avoid when solving linear equations include:

  • Not distributing the negative sign: Make sure to distribute the negative sign to the terms inside the parentheses.
  • Not combining like terms: Make sure to combine the constants and the xx terms separately.
  • Not simplifying the equation: Make sure to cancel out any terms that are the same on both sides of the equation.
  • Not checking your solution: Make sure to plug your solution back into the original equation to make sure it is true.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to solve them. By following the steps outlined in this article and avoiding common mistakes, you can master the skill of solving linear equations and apply it to many real-world applications.