Select All The Systems Of Equations That Have Exactly One Solution.(A) { Y = 3 X + 1 Y = − 3 X + 7 \left\{\begin{array}{l}y=3x+1 \\ Y=-3x+7\end{array}\right. { Y = 3 X + 1 Y = − 3 X + 7 ​ (B) { Y = 3 X + 1 Y = X + 1 \left\{\begin{array}{l}y=3x+1 \\ Y=x+1\end{array}\right. { Y = 3 X + 1 Y = X + 1 ​ (C) $\left{\begin{array}{l}y=3x+x \

Introduction to Systems of Equations

Systems of equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of multiple equations that involve multiple variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously. In this article, we will focus on selecting systems of equations that have exactly one solution.

What is a System of Equations?

A system of equations is a set of two or more equations that involve multiple variables. Each equation is a statement that two expressions are equal, and the goal is to find the values of the variables that make all the equations true. Systems of equations can be linear or nonlinear, and they can involve one or more variables.

Types of Solutions to Systems of Equations

When solving a system of equations, there are three possible outcomes:

1. One solution: The system has exactly one solution, which means that there is only one set of values that satisfies all the equations.
2. Infinitely many solutions: The system has infinitely many solutions, which means that there are many sets of values that satisfy all the equations.
3. No solution: The system has no solution, which means that there is no set of values that satisfies all the equations.

Selecting Systems of Equations with Exactly One Solution

To select systems of equations with exactly one solution, we need to consider the following:

1. Linear independence: The equations in the system must be linearly independent, which means that none of the equations can be expressed as a linear combination of the others.
2. No parallel lines: The equations in the system must not be parallel, which means that the lines represented by the equations must intersect at a single point.
3. No coincident lines: The equations in the system must not represent coincident lines, which means that the lines must not be identical.

Analyzing the Given Systems of Equations

Now, let's analyze the given systems of equations and determine which ones have exactly one solution.

System (A)

$\left\{\begin{array}{l}y=3x+1 \\ y=-3x+7\end{array}\right.$

To determine if this system has exactly one solution, we need to check if the equations are linearly independent and if the lines are not parallel.

The first equation is $y=3x+1$, and the second equation is $y=-3x+7$. We can rewrite the second equation as $y=3x-7$ by multiplying both sides by $-1$.

Now, we can see that the two equations are parallel, since they have the same slope ($3$) but different intercepts. Therefore, this system has no solution.

System (B)

$\left\{\begin{array}{l}y=3x+1 \\ y=x+1\end{array}\right.$

To determine if this system has exactly one solution, we need to check if the equations are linearly independent and if the lines are not parallel.

The first equation is $y=3x+1$, and the second equation is $y=x+1$. We can rewrite the first equation as $y=x+1$ by subtracting $3x$ from both sides.

Now, we can see that the two equations are identical, since they have the same slope ($1$) and intercept ($1$). Therefore, this system has infinitely many solutions.

System (C)

$\left\{\begin{array}{l}y=3x+x \\ y=4x\end{array}\right.$

To determine if this system has exactly one solution, we need to check if the equations are linearly independent and if the lines are not parallel.

The first equation is $y=3x+x$, which simplifies to $y=4x$. The second equation is $y=4x$.

Now, we can see that the two equations are identical, since they have the same slope ($4$) and intercept ($0$). Therefore, this system has infinitely many solutions.

Conclusion

In conclusion, only one of the given systems of equations has exactly one solution. This system is:

• System (A): $\left\{\begin{array}{l}y=3x+1 \\ y=-3x+7\end{array}\right.$

This system has no solution, since the equations are parallel.

The other two systems have infinitely many solutions, since the equations are identical or parallel.

Final Answer

The final answer is:

• System (A): $\left\{\begin{array}{l}y=3x+1 \\ y=-3x+7\end{array}\right.$
  

Introduction to Systems of Equations Q&A

In our previous article, we discussed systems of equations and how to select systems with exactly one solution. In this article, we will answer some frequently asked questions about systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve multiple variables. Each equation is a statement that two expressions are equal, and the goal is to find the values of the variables that make all the equations true.

Q: What are the different types of solutions to systems of equations?

A: There are three possible outcomes when solving a system of equations:

1. One solution: The system has exactly one solution, which means that there is only one set of values that satisfies all the equations.
2. Infinitely many solutions: The system has infinitely many solutions, which means that there are many sets of values that satisfy all the equations.
3. No solution: The system has no solution, which means that there is no set of values that satisfies all the equations.

Q: How do I determine if a system of equations has exactly one solution?

A: To determine if a system of equations has exactly one solution, you need to check if the equations are linearly independent and if the lines are not parallel. If the equations are linearly independent and the lines are not parallel, then the system has exactly one solution.

Q: What is linear independence?

A: Linear independence means that none of the equations in the system can be expressed as a linear combination of the others. In other words, each equation must be unique and cannot be derived from the other equations.

Q: What is the difference between parallel and coincident lines?

A: Parallel lines are lines that never intersect, while coincident lines are lines that are identical and intersect at every point. In the context of systems of equations, parallel lines mean that the equations have the same slope but different intercepts, while coincident lines mean that the equations have the same slope and intercept.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including:

1. Substitution method: Substitute one equation into the other equation to eliminate one variable.
2. Elimination method: Add or subtract the equations to eliminate one variable.
3. Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the importance of systems of equations in real-life applications?

A: Systems of equations have numerous real-life applications, including:

1. Physics and engineering: Systems of equations are used to model real-world problems, such as motion, forces, and energies.
2. Economics: Systems of equations are used to model economic systems, such as supply and demand, and to make predictions about economic trends.
3. Computer science: Systems of equations are used to model complex systems, such as computer networks and algorithms.

Q: Can you provide examples of systems of equations with exactly one solution?

A: Yes, here are a few examples:

1. $\left\{\begin{array}{l}y=2x+1 \\ y=2x-3\end{array}\right.$
2. $\left\{\begin{array}{l}y=x+2 \\ y=2x-1\end{array}\right.$
3. $\left\{\begin{array}{l}y=3x-2 \\ y=3x+1\end{array}\right.$

Conclusion

In conclusion, systems of equations are a fundamental concept in mathematics, and they have numerous real-life applications. By understanding the different types of solutions to systems of equations and how to determine if a system has exactly one solution, you can apply this knowledge to solve real-world problems.

Final Answer

The final answer is:

• Systems of equations are a fundamental concept in mathematics and have numerous real-life applications.

Additional Resources

For more information on systems of equations, please refer to the following resources:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "Calculus" by James Stewart
• Online resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Software: Mathematica, Maple, MATLAB