Two Systems Of Equations Are Given Below. For Each System, Choose The Best Description Of Its Solution. If Applicable, Give The Solution.System A$\[ \begin{align*} 5x - Y &= -9 \\ -5x + Y &= 9 \end{align*} \\]- The System Has No Solution.- The

Introduction

Systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore two systems of equations and provide the best description of their solutions. We will also give the solution if applicable.

System A

The first system of equations is given by:

$$\begin{align*} 5x - y &= -9 \\ -5x + y &= 9 \end{align*}$$

To solve this system, we can use the method of substitution or elimination. Let's use the elimination method.

Step 1: Multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same

We can multiply the first equation by 1 and the second equation by 1. This will give us:

$$\begin{align*} 5x - y &= -9 \\ -5x + y &= 9 \end{align*}$$

Step 2: Add both equations to eliminate the variable y

Now, we can add both equations to eliminate the variable y:

$$\begin{align*} (5x - y) + (-5x + y) &= -9 + 9 \\ 0 &= 0 \end{align*}$$

As we can see, the resulting equation is a contradiction, which means that the system has no solution.

Conclusion

The system of equations given by:

$$\begin{align*} 5x - y &= -9 \\ -5x + y &= 9 \end{align*}$$

has no solution.

System B

The second system of equations is given by:

$$\begin{align*} x + y &= 4 \\ x - y &= 2 \end{align*}$$

To solve this system, we can use the method of substitution or elimination. Let's use the elimination method.

Step 1: Multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same

We can multiply the first equation by 1 and the second equation by 1. This will give us:

$$\begin{align*} x + y &= 4 \\ x - y &= 2 \end{align*}$$

Step 2: Add both equations to eliminate the variable y

Now, we can add both equations to eliminate the variable y:

$$\begin{align*} (x + y) + (x - y) &= 4 + 2 \\ 2x &= 6 \end{align*}$$

Step 3: Solve for x

Now, we can solve for x:

$$\begin{align*} 2x &= 6 \\ x &= 3 \end{align*}$$

Step 4: Substitute x into one of the original equations to solve for y

Now, we can substitute x into one of the original equations to solve for y:

$$\begin{align*} x + y &= 4 \\ 3 + y &= 4 \\ y &= 1 \end{align*}$$

Conclusion

The system of equations given by:

$$\begin{align*} x + y &= 4 \\ x - y &= 2 \end{align*}$$

has a solution, which is x = 3 and y = 1.

Conclusion

In this article, we have explored two systems of equations and provided the best description of their solutions. We have shown that the first system has no solution, while the second system has a solution, which is x = 3 and y = 1. We have also provided a step-by-step guide on how to solve systems of equations using the elimination method.

Tips and Tricks

• When solving systems of equations, it's essential to choose the correct method, such as substitution or elimination.
• When using the elimination method, make sure to multiply the equations by necessary multiples such that the coefficients of the variables are the same.
• When adding or subtracting equations, make sure to combine like terms.
• When solving for variables, make sure to isolate the variable on one side of the equation.

Common Mistakes

• Not choosing the correct method for solving the system.
• Not multiplying the equations by necessary multiples.
• Not combining like terms when adding or subtracting equations.
• Not isolating the variable on one side of the equation.

Real-World Applications

Systems of equations have numerous real-world applications, such as:

• Physics and Engineering: Systems of equations are used to model real-world problems, such as motion, forces, and energies.
• Economics: Systems of equations are used to model economic systems, such as supply and demand, and to make predictions about economic trends.
• Computer Science: Systems of equations are used in computer science to solve problems, such as linear programming and optimization.

Conclusion

In conclusion, systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. We have explored two systems of equations and provided the best description of their solutions. We have also provided a step-by-step guide on how to solve systems of equations using the elimination method.