Which One Of The Following Equations Is Equivalent To $3x - 12 = -9$?A. $x - 5 = -6$ B. $ 7 X − 3 = 4 7x - 3 = 4 7 X − 3 = 4 [/tex] C. $2x = 4$ D. $x + 3 = X - 1$
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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 the concept of linear equations, how to solve them, and provide a step-by-step guide on how to identify equivalent equations.
What are Linear Equations?
A linear equation is an equation in which the highest power of the variable(s) 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. Linear equations can be solved using various methods, including addition, subtraction, multiplication, and division.
Solving Linear Equations
To solve a linear equation, we need to isolate the variable (x) on one side of the equation. We can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Example 1: Solving the Equation 3x - 12 = -9
Let's solve the equation 3x - 12 = -9 using the addition method.
Step 1: Add 12 to both sides of the equation to get rid of the negative term.
3x - 12 + 12 = -9 + 12
This simplifies to:
3x = 3
Step 2: Divide both sides of the equation by 3 to solve for x.
3x / 3 = 3 / 3
This simplifies to:
x = 1
Therefore, the solution to the equation 3x - 12 = -9 is x = 1.
Equivalent Equations
Two equations are said to be equivalent if they have the same solution. In other words, if we substitute the value of x from one equation into the other equation, the two equations will be true.
Example 2: Finding Equivalent Equations
Let's find an equivalent equation to 3x - 12 = -9.
Step 1: Add 12 to both sides of the equation to get rid of the negative term.
3x - 12 + 12 = -9 + 12
This simplifies to:
3x = 3
Step 2: Divide both sides of the equation by 3 to solve for x.
3x / 3 = 3 / 3
This simplifies to:
x = 1
Now, let's find an equivalent equation to 3x - 12 = -9. We can do this by multiplying both sides of the equation by 2.
2(3x - 12) = 2(-9)
This simplifies to:
6x - 24 = -18
Now, let's add 24 to both sides of the equation to get rid of the negative term.
6x - 24 + 24 = -18 + 24
This simplifies to:
6x = 6
Step 3: Divide both sides of the equation by 6 to solve for x.
6x / 6 = 6 / 6
This simplifies to:
x = 1
Therefore, the equivalent equation to 3x - 12 = -9 is 6x = 6.
Which One of the Following Equations is Equivalent to 3x - 12 = -9?
Now that we have learned how to solve linear equations and find equivalent equations, let's apply this knowledge to the given problem.
The given equation is 3x - 12 = -9. We need to find an equivalent equation to this equation.
Option A: x - 5 = -6
Let's solve this equation to see if it's equivalent to 3x - 12 = -9.
Step 1: Add 5 to both sides of the equation to get rid of the negative term.
x - 5 + 5 = -6 + 5
This simplifies to:
x = -1
Step 2: Multiply both sides of the equation by 3 to see if it's equivalent to 3x - 12 = -9.
3x = -3
This is not equivalent to 3x - 12 = -9.
Option B: 7x - 3 = 4
Let's solve this equation to see if it's equivalent to 3x - 12 = -9.
Step 1: Add 3 to both sides of the equation to get rid of the negative term.
7x - 3 + 3 = 4 + 3
This simplifies to:
7x = 7
Step 2: Divide both sides of the equation by 7 to see if it's equivalent to 3x - 12 = -9.
x = 1
This is equivalent to 3x - 12 = -9.
Option C: 2x = 4
Let's solve this equation to see if it's equivalent to 3x - 12 = -9.
Step 1: Divide both sides of the equation by 2 to solve for x.
2x / 2 = 4 / 2
This simplifies to:
x = 2
This is not equivalent to 3x - 12 = -9.
Option D: x + 3 = x - 1
Let's solve this equation to see if it's equivalent to 3x - 12 = -9.
Step 1: Add 1 to both sides of the equation to get rid of the negative term.
x + 3 + 1 = x - 1 + 1
This simplifies to:
x + 4 = x
Step 2: Subtract x from both sides of the equation to solve for x.
x + 4 - x = x - x
This simplifies to:
4 = 0
This is not equivalent to 3x - 12 = -9.
Therefore, the correct answer is Option B: 7x - 3 = 4.
Conclusion
In this article, we have learned how to solve linear equations and find equivalent equations. We have also applied this knowledge to the given problem and found that the correct answer is Option B: 7x - 3 = 4.
We hope that this article has provided you with a clear understanding of how to solve linear equations and find equivalent equations. If you have any questions or need further clarification, please don't hesitate to ask.
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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 of 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 need to isolate the variable (x) 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 an equivalent equation?
A: Two equations are said to be equivalent if they have the same solution. In other words, if we substitute the value of x from one equation into the other equation, the two equations will be true.
Q: How do I find an equivalent equation?
A: To find an equivalent equation, you can multiply or divide both sides of the equation by the same value. You can also add or subtract the same value from both sides of the equation.
Q: Can I have multiple equivalent equations?
A: Yes, you can have multiple equivalent equations. For example, if you have the equation 3x - 12 = -9, you can also have the equivalent equations 6x = 6, 9x - 27 = -27, and so on.
Q: How do I know if an equation is equivalent to another equation?
A: To know if an equation is equivalent to another equation, you need to check if they have the same solution. You can do this by substituting the value of x from one equation into the other equation and checking if the two equations are true.
Q: Can I use equivalent equations to solve a system of equations?
A: Yes, you can use equivalent equations to solve a system of equations. By finding equivalent equations, you can simplify the system of equations and make it easier to solve.
Q: What are some common mistakes to avoid when working with linear equations and equivalent equations?
A: Some common mistakes to avoid when working with linear equations and equivalent equations include:
- Not isolating the variable (x) on one side of the equation
- Not checking if the equation is true before substituting the value of x
- Not using the correct method to find an equivalent equation
- Not checking if the equivalent equation has the same solution as the original equation
Q: How can I practice solving linear equations and finding equivalent equations?
A: You can practice solving linear equations and finding equivalent equations by working through practice problems and exercises. You can also use online resources and tools to help you practice and learn.
Q: What are some real-world applications of linear equations and equivalent equations?
A: Linear equations and equivalent equations have many real-world applications, including:
- Physics and engineering: Linear equations are used to describe the motion of objects and the behavior of physical systems.
- Economics: Linear equations are used to model economic systems and make predictions about economic trends.
- Computer science: Linear equations are used in computer algorithms and data analysis.
- Statistics: Linear equations are used to analyze and interpret data.
Q: Can I use linear equations and equivalent equations to solve problems in other areas of mathematics?
A: Yes, you can use linear equations and equivalent equations to solve problems in other areas of mathematics, including algebra, geometry, and calculus. By mastering linear equations and equivalent equations, you can develop a strong foundation in mathematics and apply it to a wide range of problems and applications.