Is ( 3 , − 5 (3, -5 ( 3 , − 5 ] A Solution To This System Of Equations? \begin{align*} y &= -\frac{1}{3}x - 4 \\ y &= -2x + 1 \end{align*} A. Yes B. No

Understanding the Problem

To determine if the point $(3, -5)$ is a solution to the given system of equations, we need to substitute the coordinates into each equation and check if the resulting statement is true. The system consists of two linear equations in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

The First Equation: $y = -\frac{1}{3}x - 4$

The first equation represents a line with a slope of $-\frac{1}{3}$ and a y-intercept of $-4$. To check if the point $(3, -5)$ satisfies this equation, we substitute $x = 3$ and $y = -5$ into the equation.

y = -\frac{1}{3}x - 4
-\frac{1}{3}(3) - 4 = -1 - 4 = -5

The Second Equation: $y = -2x + 1$

The second equation represents a line with a slope of $-2$ and a y-intercept of $1$. To check if the point $(3, -5)$ satisfies this equation, we substitute $x = 3$ and $y = -5$ into the equation.

y = -2x + 1
-2(3) + 1 = -6 + 1 = -5

Analyzing the Results

From the previous section, we can see that the point $(3, -5)$ satisfies both equations, as the resulting statements are true. This means that the point $(3, -5)$ lies on both lines represented by the equations.

Conclusion

Based on the analysis, we can conclude that the point $(3, -5)$ is indeed a solution to the given system of equations. This is because the point satisfies both equations, indicating that it lies on both lines.

Why is this Important?

Understanding whether a point is a solution to a system of equations is crucial in various mathematical and real-world applications. For instance, in physics, it can help determine the position of an object in a two-dimensional space. In computer science, it can aid in solving problems involving linear equations and inequalities.

Real-World Applications

The concept of solving systems of equations has numerous real-world applications, including:

• Physics and Engineering: Solving systems of equations is essential in physics and engineering to determine the position, velocity, and acceleration of objects in two-dimensional and three-dimensional spaces.
• Computer Science: Solving systems of equations is used in computer science to solve problems involving linear equations and inequalities, such as linear programming and optimization.
• Economics: Solving systems of equations is used in economics to model and analyze economic systems, including supply and demand curves.

Tips and Tricks

When solving systems of equations, it's essential to:

• Substitute values carefully: When substituting values into an equation, ensure that the correct values are used to avoid errors.
• Check for extraneous solutions: When solving a system of equations, check for extraneous solutions that may arise from the substitution process.
• Use graphing tools: Graphing tools can help visualize the solution to a system of equations and identify any extraneous solutions.

Conclusion

In conclusion, the point $(3, -5)$ is indeed a solution to the given system of equations. This is because the point satisfies both equations, indicating that it lies on both lines. Understanding whether a point is a solution to a system of equations is crucial in various mathematical and real-world applications. By following the tips and tricks outlined in this article, you can effectively solve systems of equations and apply the concepts to real-world problems.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are related to each other through a common variable or variables. In this article, we are dealing with a system of two linear equations in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I determine if a point is a solution to a system of equations?

To determine if a point is a solution to a system of equations, you need to substitute the coordinates into each equation and check if the resulting statement is true. If the point satisfies both equations, then it is a solution to the system.

Q: What is the difference between a solution and an extraneous solution?

A solution is a point that satisfies both equations in a system of equations. An extraneous solution, on the other hand, is a point that satisfies one or both equations but is not a valid solution to the system.

Q: How do I identify extraneous solutions?

Extraneous solutions can arise from the substitution process. To identify extraneous solutions, you need to check if the point satisfies both equations. If it does not satisfy one or both equations, then it is an extraneous solution.

Q: What are some real-world applications of solving systems of equations?

Solving systems of equations has numerous real-world applications, including physics, engineering, computer science, and economics. In physics, it can help determine the position, velocity, and acceleration of objects in two-dimensional and three-dimensional spaces. In computer science, it can aid in solving problems involving linear equations and inequalities.

Q: What are some tips and tricks for solving systems of equations?

Some tips and tricks for solving systems of equations include:

• Substitute values carefully: When substituting values into an equation, ensure that the correct values are used to avoid errors.
• Check for extraneous solutions: When solving a system of equations, check for extraneous solutions that may arise from the substitution process.
• Use graphing tools: Graphing tools can help visualize the solution to a system of equations and identify any extraneous solutions.

Q: How do I graph a system of equations?

To graph a system of equations, you can use graphing tools or plot the lines represented by the equations on a coordinate plane. The point of intersection of the two lines represents the solution to the system.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A system of linear equations consists of two or more linear equations in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. A system of nonlinear equations, on the other hand, consists of two or more nonlinear equations that cannot be written in the form of $y = mx + b$.

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

Solving a system of nonlinear equations can be more complex than solving a system of linear equations. There are various methods for solving systems of nonlinear equations, including substitution, elimination, and graphing.

Q: What are some common mistakes to avoid when solving systems of equations?

Some common mistakes to avoid when solving systems of equations include:

• Substituting incorrect values: When substituting values into an equation, ensure that the correct values are used to avoid errors.
• Failing to check for extraneous solutions: When solving a system of equations, check for extraneous solutions that may arise from the substitution process.
• Not using graphing tools: Graphing tools can help visualize the solution to a system of equations and identify any extraneous solutions.

Q: How do I choose the best method for solving a system of equations?

The best method for solving a system of equations depends on the specific problem and the type of equations involved. Some common methods include substitution, elimination, and graphing. Choose the method that best suits the problem and the type of equations involved.