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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Understanding the Problem
When solving linear equations, it's essential to understand the properties of equality and how to manipulate expressions to isolate the variable. In this article, we'll explore how to solve linear equations and apply this knowledge to a specific problem.
What is a Linear Equation?
A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:
ax + b = c
where a, b, and c are constants, and x is the variable.
Properties of Equality
When solving linear equations, we use the properties of equality to manipulate the expressions. The two main properties of equality are:
- Addition Property: If a = b, then a + c = b + c.
- Multiplication Property: If a = b, then ac = bc.
Solving the Equation
Now, let's apply these properties to the given equation:
To solve this equation, we'll start by simplifying both sides using the distributive property.
Distributive Property
The distributive property states that for any numbers a, b, and c:
a(b + c) = ab + ac
We can apply this property to both sides of the equation:
Combining Like Terms
Now, let's combine like terms on both sides of the equation:
Simplifying the Equation
We can simplify the equation by canceling out the like terms:
This equation is true for all values of x, which means that the equation has an infinite number of solutions.
Conclusion
In conclusion, when solving linear equations, it's essential to understand the properties of equality and how to manipulate expressions to isolate the variable. By applying these properties to the given equation, we found that the equation has an infinite number of solutions.
Answer
The correct answer is:
C. The equation has no solution.
However, this is not entirely accurate. The equation has an infinite number of solutions, not no solution. But since the options do not include "the equation has an infinite number of solutions," we must choose the closest answer.
Final Thoughts
Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the properties of equality and how to manipulate expressions to isolate the variable. By applying these properties to the given equation, we found that the equation has an infinite number of solutions. This demonstrates the importance of careful analysis and attention to detail when solving mathematical problems.
Additional Resources
For more information on solving linear equations, check out the following resources:
- Khan Academy: Solving Linear Equations
- Mathway: Solving Linear Equations
- Wolfram Alpha: Solving Linear Equations
These resources provide a comprehensive overview of solving linear equations and offer additional practice problems to help you improve your skills.
Conclusion
In conclusion, solving linear equations is a critical concept in mathematics that requires a deep understanding of the properties of equality and how to manipulate expressions to isolate the variable. By applying these properties to the given equation, we found that the equation has an infinite number of solutions. This demonstrates the importance of careful analysis and attention to detail when solving mathematical problems.
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Understanding the Problem
When solving linear equations, it's essential to understand the properties of equality and how to manipulate expressions to isolate the variable. In this article, we'll explore some common questions and answers related to 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(s) is 1. It can be written in the form:
ax + b = c
where a, b, and c are constants, and x is the variable.
Q: What are the properties of equality?
A: The two main properties of equality are:
- Addition Property: If a = b, then a + c = b + c.
- Multiplication Property: If a = b, then ac = bc.
Q: How do I simplify an equation using the distributive property?
A: To simplify an equation using the distributive property, you can apply the following steps:
- Distribute the coefficients to the terms inside the parentheses.
- Combine like terms on both sides of the 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. A quadratic equation, on the other hand, is an equation in which the highest power of the variable(s) is 2. For example:
Linear Equation: 2x + 3 = 5
Quadratic Equation: x^2 + 4x + 4 = 0
Q: How do I solve a linear equation with multiple variables?
A: To solve a linear equation with multiple variables, you can use the following steps:
- Isolate one of the variables by adding or subtracting the same value to both sides of the equation.
- Use the properties of equality to simplify the equation.
- Repeat the process until you have isolated all the variables.
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 or more variables. A system of linear equations, on the other hand, is a set of two or more linear equations with the same variables. For example:
Linear Equation: 2x + 3 = 5
System of Linear Equations:
2x + 3 = 5 x - 2 = 3
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you can use the following steps:
- Use the substitution method to substitute one equation into the other.
- Use the elimination method to eliminate one of the variables.
- Solve for the remaining variable.
Conclusion
In conclusion, solving linear equations is a critical concept in mathematics that requires a deep understanding of the properties of equality and how to manipulate expressions to isolate the variable. By applying these properties to various equations, we can solve for the unknown variable and understand the relationships between the variables.
Additional Resources
For more information on solving linear equations, check out the following resources:
- Khan Academy: Solving Linear Equations
- Mathway: Solving Linear Equations
- Wolfram Alpha: Solving Linear Equations
These resources provide a comprehensive overview of solving linear equations and offer additional practice problems to help you improve your skills.
Final Thoughts
Solving linear equations is a fundamental concept in mathematics that requires a deep understanding of the properties of equality and how to manipulate expressions to isolate the variable. By applying these properties to various equations, we can solve for the unknown variable and understand the relationships between the variables. With practice and patience, you can become proficient in solving linear equations and tackle more complex mathematical problems.