What Is The Solution To The System Of Equations?${ \begin{array}{l} y = \frac{2}{3}x + 3 \ x = -2 \end{array} }$A. { (-2, -\frac{15}{2})$}$B. { (-2, \frac{5}{3})$}$C. { (-2, \frac{11}{6})$} D . \[ D. \[ D . \[ (-2,

Introduction

When dealing with systems of equations, it's essential to understand how to find the solution that satisfies both equations. In this article, we will explore a system of equations and determine the solution using a step-by-step approach.

Understanding the System of Equations

The given system of equations consists of two equations:

1. $y = \frac{2}{3}x + 3$
2. $x = -2$

The first equation is a linear equation in the slope-intercept form, where the slope is $\frac{2}{3}$ and the y-intercept is 3. The second equation is a simple linear equation where the value of x is -2.

Finding the Solution

To find the solution to the system of equations, we can substitute the value of x from the second equation into the first equation.

Substituting x into the First Equation

We will substitute x = -2 into the first equation:

$y = \frac{2}{3}(-2) + 3$

Simplifying the Equation

Now, we will simplify the equation by multiplying $\frac{2}{3}$ with -2 and then adding 3:

$y = -\frac{4}{3} + 3$

Combining Like Terms

Next, we will combine the like terms by converting 3 into a fraction with a denominator of 3:

$y = -\frac{4}{3} + \frac{9}{3}$

Simplifying the Fraction

Now, we will simplify the fraction by adding the numerators:

$y = \frac{5}{3}$

Conclusion

The solution to the system of equations is $(-2, \frac{5}{3})$. This means that when x is -2, the value of y is $\frac{5}{3}$.

Discussion

The correct answer is B. $(-2, \frac{5}{3})$. This is because when x is -2, the value of y is $\frac{5}{3}$, which is the solution to the system of equations.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Substitute x = -2 into the first equation: $y = \frac{2}{3}(-2) + 3$
2. Simplify the equation: $y = -\frac{4}{3} + 3$
3. Combine like terms: $y = -\frac{4}{3} + \frac{9}{3}$
4. Simplify the fraction: $y = \frac{5}{3}$

Final Answer

The final answer is B. $(-2, \frac{5}{3})$.

Introduction

Systems of equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields, including science, engineering, and economics. In this article, we will address some of the most 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 are related to each other through a common variable or variables. The goal is to find the values of the variables that satisfy all the equations in the system.

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

A: There are two main types of systems of equations:

1. Linear systems of equations: These are systems of equations where each equation is a linear equation, meaning it can be written in the form ax + by = c, where a, b, and c are constants.
2. Non-linear systems of equations: These are systems of equations where at least one equation is not a linear equation.

Q: How do I solve a system of equations?

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

1. Substitution method: This involves substituting the value of one variable from one equation into the other equation.
2. Elimination method: This involves eliminating one variable by adding or subtracting the equations.
3. Graphical method: This involves graphing the equations on a coordinate plane and finding the point of intersection.
4. Matrix method: This involves using matrices to solve the system of equations.

Q: What is the difference between a system of equations and a single equation?

A: A single equation is a statement that expresses a relationship between two or more variables, while a system of equations is a set of two or more equations that are related to each other through a common variable or variables.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution if the equations are inconsistent, meaning that they cannot be true at the same time.

Q: Can a system of equations have an infinite number of solutions?

A: Yes, a system of equations can have an infinite number of solutions if the equations are dependent, meaning that they are essentially the same equation.

Q: How do I determine if a system of equations has a unique solution, no solution, or an infinite number of solutions?

A: To determine the number of solutions, you can use the following methods:

1. Graphical method: Graph the equations on a coordinate plane and observe the number of points of intersection.
2. Matrix method: Use the matrix method to find the determinant of the coefficient matrix. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system has either no solution or an infinite number of solutions.
3. Substitution method: Substitute the value of one variable from one equation into the other equation. If the resulting equation is true, the system has a unique solution. If the resulting equation is false, the system has no solution.

Conclusion

Systems of equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields. By answering some of the most frequently asked questions about systems of equations, we hope to have provided a better understanding of this important topic.

Additional Resources

For more information on systems of equations, we recommend the following resources:

• Khan Academy: Systems of Equations
• Mathway: Systems of Equations
• Wolfram Alpha: Systems of Equations

Final Answer

The final answer is that systems of equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields. By using the methods and resources provided in this article, you can gain a better understanding of systems of equations and improve your problem-solving skills.