Which Equation Can Pair With $3x + 4y = 8$ To Create A Consistent And Independent System?A. $6x + 8y = 16$B. $ − 3 X − 4 Y = − 6 -3x - 4y = -6 − 3 X − 4 Y = − 6 [/tex]C. $6x - 3y = 2$D. $-3x - 4y = -8$

by ADMIN 210 views

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve variables raised to the power of 1. These equations can be solved using various methods, including substitution, elimination, and graphing. However, for a system to be considered consistent and independent, it must have a unique solution. In this article, we will explore which equation can pair with the given equation $3x + 4y = 8$ to create a consistent and independent system.

What is a Consistent and Independent System?

A consistent system of linear equations is one that has at least one solution. On the other hand, an independent system has a unique solution. In other words, the equations in an independent system are not multiples of each other. To create a consistent and independent system, we need to find an equation that is not a multiple of the given equation.

The Given Equation

The given equation is $3x + 4y = 8$. To create a consistent and independent system, we need to find an equation that is not a multiple of this equation. Let's analyze the options provided.

Option A: $6x + 8y = 16$

This equation is a multiple of the given equation. If we multiply the given equation by 2, we get $6x + 8y = 16$. Therefore, this option does not create a consistent and independent system.

Option B: $-3x - 4y = -6$

This equation is also a multiple of the given equation. If we multiply the given equation by -1, we get $-3x - 4y = -6$. Therefore, this option does not create a consistent and independent system.

Option C: $6x - 3y = 2$

This equation is not a multiple of the given equation. If we multiply the given equation by 2 and then subtract the given equation from it, we get $6x - 3y = 2$. Therefore, this option creates a consistent and independent system.

Option D: $-3x - 4y = -8$

This equation is a multiple of the given equation. If we multiply the given equation by -1 and then add 2 to both sides, we get $-3x - 4y = -8$. Therefore, this option does not create a consistent and independent system.

Conclusion

In conclusion, the equation that can pair with $3x + 4y = 8$ to create a consistent and independent system is Option C: $6x - 3y = 2$. This equation is not a multiple of the given equation and can be used to create a unique solution.

Why is Option C the Correct Answer?

Option C is the correct answer because it is not a multiple of the given equation. When we multiply the given equation by 2, we get $6x + 8y = 16$. However, Option C has a different coefficient for the y-variable, which is -3. This means that the two equations are not multiples of each other and can be used to create a consistent and independent system.

How to Solve the System?

To solve the system, we can use the substitution or elimination method. Let's use the elimination method. We can multiply the given equation by 2 and then subtract the given equation from it to get $6x - 3y = 2$. This equation is the same as Option C. We can then solve for x and y using the two equations.

Step 1: Multiply the Given Equation by 2

6x+8y=166x + 8y = 16

Step 2: Subtract the Given Equation from the Resulting Equation

6x+8y−(3x+4y)=16−86x + 8y - (3x + 4y) = 16 - 8

3x+4y=83x + 4y = 8

6x−3y=26x - 3y = 2

Step 3: Solve for x and y

We can solve for x and y using the two equations. Let's solve for x first.

3x+4y=83x + 4y = 8

6x−3y=26x - 3y = 2

We can multiply the first equation by 2 and then add the second equation to it to get:

6x+8y=166x + 8y = 16

6x−3y=26x - 3y = 2

6x+8y+6x−3y=16+26x + 8y + 6x - 3y = 16 + 2

12x+5y=1812x + 5y = 18

We can then solve for x and y using this equation.

Step 4: Solve for x and y

We can solve for x and y using the equation $12x + 5y = 18$. Let's solve for x first.

12x+5y=1812x + 5y = 18

5y=18−12x5y = 18 - 12x

y=18−12x5y = \frac{18 - 12x}{5}

We can then substitute this expression for y into one of the original equations to solve for x.

Step 5: Substitute the Expression for y into One of the Original Equations

We can substitute the expression for y into the first original equation to solve for x.

3x+4y=83x + 4y = 8

3x+4(18−12x5)=83x + 4\left(\frac{18 - 12x}{5}\right) = 8

3x+72−48x5=83x + \frac{72 - 48x}{5} = 8

15x+72−48x=4015x + 72 - 48x = 40

−33x=−32-33x = -32

x=−32−33x = \frac{-32}{-33}

x=3233x = \frac{32}{33}

We can then substitute this value of x into one of the original equations to solve for y.

Step 6: Substitute the Value of x into One of the Original Equations

We can substitute the value of x into the first original equation to solve for y.

3x+4y=83x + 4y = 8

3(3233)+4y=83\left(\frac{32}{33}\right) + 4y = 8

9633+4y=8\frac{96}{33} + 4y = 8

4y=8−96334y = 8 - \frac{96}{33}

4y=264−96334y = \frac{264 - 96}{33}

4y=168334y = \frac{168}{33}

y=16833×4y = \frac{168}{33 \times 4}

y=4233y = \frac{42}{33}

y=1411y = \frac{14}{11}

Therefore, the solution to the system is $x = \frac{32}{33}$ and $y = \frac{14}{11}$.

Conclusion

Q: What is a consistent and independent system of linear equations?

A: A consistent system of linear equations is one that has at least one solution. On the other hand, an independent system has a unique solution. In other words, the equations in an independent system are not multiples of each other.

Q: How do I determine if a system of linear equations is consistent and independent?

A: To determine if a system of linear equations is consistent and independent, you need to check if the equations are not multiples of each other. You can do this by multiplying one equation by a constant and then subtracting the other equation from it. If the resulting equation is not a multiple of the other equation, then the system is consistent and independent.

Q: What is the difference between a consistent and independent system and a consistent and dependent system?

A: A consistent and dependent system is one where the equations are multiples of each other, but the system still has a solution. On the other hand, a consistent and independent system is one where the equations are not multiples of each other and the system has a unique solution.

Q: How do I solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including the substitution method, elimination method, and graphing method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves multiplying one or both equations by a constant and then adding or subtracting the equations to eliminate one variable. The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of linear equations by solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of linear equations by multiplying one or both equations by a constant and then adding or subtracting the equations to eliminate one variable.

Q: What is the graphing method?

A: The graphing method is a method of solving a system of linear equations by graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find two points on the line and then draw a line through those points. You can find two points on the line by substituting different values of x into the equation and solving for y.

Q: What is the point of intersection?

A: The point of intersection is the point where the two lines intersect. This is the solution to the system of linear equations.

Q: How do I find the point of intersection?

A: To find the point of intersection, you need to graph the two lines on a coordinate plane and find the point where they intersect.

Q: What is the solution to a system of linear equations?

A: The solution to a system of linear equations is the point where the two lines intersect. This is the value of x and y that satisfies both equations.

Q: How do I check if my solution is correct?

A: To check if your solution is correct, you need to substitute the values of x and y into both equations and make sure that both equations are true.

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

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

  • Not checking if the equations are consistent and independent
  • Not using the correct method to solve the system
  • Not checking if the solution is correct
  • Not being careful when multiplying and dividing equations

Q: How can I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own and then checking your answers with a calculator or online tool.