Creating A System Of Equivalent EquationsConsider The Equation $2y - 4x = 12$.Which Equation, When Graphed With The Given Equation, Will Form A System With One Solution?A. $-y - 2x = 6$ B. $-y + 2x = 12$ C. $y = 2x +

Feb 26, 2025 by ADMIN 219 views

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Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve the same variables. When we have a system of linear equations, we can use various methods to solve for the values of the variables. One of the methods is to create a system of equivalent equations, which is a set of equations that have the same solution. In this article, we will discuss how to create a system of equivalent equations and provide examples to illustrate the concept.

What are Equivalent Equations?

Equivalent equations are equations that have the same solution. In other words, if we have two equations, and they have the same solution, then they are equivalent equations. For example, consider the equations $2y - 4x = 12$ and $y - 2x = 6$ . These two equations are equivalent because they have the same solution.

Creating a System of Equivalent Equations

To create a system of equivalent equations, we need to find an equation that has the same solution as the given equation. We can do this by manipulating the given equation using algebraic operations such as addition, subtraction, multiplication, and division.

Example 1

Consider the equation $2y - 4x = 12$ . To create a system of equivalent equations, we can add $4x$ to both sides of the equation to get $2y = 12 + 4x$ . Then, we can divide both sides of the equation by 2 to get $y = 6 + 2x$ . This is an equivalent equation to the original equation.

Example 2

Consider the equation $2y - 4x = 12$ . To create a system of equivalent equations, we can subtract $2y$ from both sides of the equation to get $-4x = 12 - 2y$ . Then, we can divide both sides of the equation by -4 to get $x = -3 + \frac{1}{2}y$ . This is an equivalent equation to the original equation.

Which Equation Will Form a System with One Solution?

Now, let's consider the equation $2y - 4x = 12$ and the three options given:

A. $-y - 2x = 6$ B. $-y + 2x = 12$ C. $y = 2x + 3$

To determine which equation will form a system with one solution, we need to check if the two equations are equivalent. If they are equivalent, then they will have the same solution.

Option A

Let's check if the equation $-y - 2x = 6$ is equivalent to the equation $2y - 4x = 12$ . We can add $y$ to both sides of the equation to get $-2x = 6 + y$ . Then, we can divide both sides of the equation by -2 to get $x = -3 - \frac{1}{2}y$ . This is not an equivalent equation to the original equation.

Option B

Let's check if the equation $-y + 2x = 12$ is equivalent to the equation $2y - 4x = 12$ . We can add $y$ to both sides of the equation to get $2x = 12 + y$ . Then, we can divide both sides of the equation by 2 to get $x = 6 + \frac{1}{2}y$ . This is not an equivalent equation to the original equation.

Option C

Let's check if the equation $y = 2x + 3$ is equivalent to the equation $2y - 4x = 12$ . We can multiply both sides of the equation by 2 to get $2y = 4x + 6$ . Then, we can add $4x$ to both sides of the equation to get $2y - 4x = 6$ . This is an equivalent equation to the original equation.

Conclusion

In conclusion, the equation $y = 2x + 3$ is the only option that will form a system with one solution when graphed with the given equation $2y - 4x = 12$ . This is because the two equations are equivalent, and they have the same solution.

Final Answer

The final answer is C.

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Introduction

In our previous article, we discussed how to create a system of equivalent equations and provided examples to illustrate the concept. In this article, we will answer some frequently asked questions about creating a system of equivalent equations.

Q&A

Q: What is the difference between equivalent equations and parallel equations?

A: Equivalent equations are equations that have the same solution, while parallel equations are equations that have no solution in common. In other words, equivalent equations have the same solution, while parallel equations have different solutions.

Q: How do I determine if two equations are equivalent?

A: To determine if two equations are equivalent, you can use the following steps:

Check if the equations have the same variables.
Check if the equations have the same coefficients for each variable.
Check if the equations have the same constant term.
If all of the above conditions are met, then the equations are equivalent.

Q: Can I create a system of equivalent equations by adding or subtracting the same value to both sides of an equation?

A: Yes, you can create a system of equivalent equations by adding or subtracting the same value to both sides of an equation. For example, if you have the equation $2y - 4x = 12$ , you can add $4x$ to both sides of the equation to get $2y = 12 + 4x$ . Then, you can divide both sides of the equation by 2 to get $y = 6 + 2x$ .

Q: Can I create a system of equivalent equations by multiplying or dividing both sides of an equation by the same non-zero value?

A: Yes, you can create a system of equivalent equations by multiplying or dividing both sides of an equation by the same non-zero value. For example, if you have the equation $2y - 4x = 12$ , you can multiply both sides of the equation by 2 to get $4y - 8x = 24$ . Then, you can divide both sides of the equation by 4 to get $y - 2x = 6$ .

Q: Can I create a system of equivalent equations by adding or subtracting the same value to both sides of an equation and then multiplying or dividing both sides of the equation by the same non-zero value?

A: Yes, you can create a system of equivalent equations by adding or subtracting the same value to both sides of an equation and then multiplying or dividing both sides of the equation by the same non-zero value. For example, if you have the equation $2y - 4x = 12$ , you can add $4x$ to both sides of the equation to get $2y = 12 + 4x$ . Then, you can multiply both sides of the equation by 2 to get $4y = 24 + 8x$ . Finally, you can divide both sides of the equation by 4 to get $y = 6 + 2x$ .

Q: Can I create a system of equivalent equations by using a combination of addition, subtraction, multiplication, and division operations?

A: Yes, you can create a system of equivalent equations by using a combination of addition, subtraction, multiplication, and division operations. For example, if you have the equation $2y - 4x = 12$ , you can add $4x$ to both sides of the equation to get $2y = 12 + 4x$ . Then, you can multiply both sides of the equation by 2 to get $4y = 24 + 8x$ . Finally, you can divide both sides of the equation by 4 to get $y = 6 + 2x$ .

Conclusion

In conclusion, creating a system of equivalent equations is a powerful tool in mathematics that can be used to solve a wide range of problems. By understanding how to create a system of equivalent equations, you can solve equations that would otherwise be difficult or impossible to solve.

Final Answer

The final answer is that creating a system of equivalent equations is a powerful tool in mathematics that can be used to solve a wide range of problems.