Is ( − 2 , − 3 (-2,-3 ( − 2 , − 3 ] A Solution To This System Of Equations? Y = 2 X + 1 Y = X − 1 \begin{array}{l} Y = 2x + 1 \\ Y = X - 1 \end{array} Y = 2 X + 1 Y = X − 1 ​ A. Yes B. No

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Introduction

When dealing with systems of linear equations, it's essential to determine whether a given point satisfies both equations. In this case, we're asked to determine if the point $(-2,-3)$ is a solution to the system of equations:

$\begin{array}{l} y = 2x + 1 \\ y = x - 1 \end{array}$

To answer this question, we'll substitute the coordinates of the point into each equation and check if the resulting statement is true.

Understanding the Equations

The first equation is $y = 2x + 1$. This is a linear equation in slope-intercept form, where the slope is 2 and the y-intercept is 1. The second equation is $y = x - 1$, which is also a linear equation in slope-intercept form, with a slope of 1 and a y-intercept of -1.

Substituting the Point into the Equations

To determine if the point $(-2,-3)$ is a solution to the system of equations, we'll substitute the x-coordinate and y-coordinate into each equation.

Substituting into the First Equation

For the first equation, $y = 2x + 1$, we'll substitute $x = -2$ and $y = -3$:

$-3 = 2(-2) + 1$

Expanding the equation, we get:

$-3 = -4 + 1$

Simplifying the equation, we get:

$-3 = -3$

This statement is true, which means that the point $(-2,-3)$ satisfies the first equation.

Substituting into the Second Equation

For the second equation, $y = x - 1$, we'll substitute $x = -2$ and $y = -3$:

$-3 = (-2) - 1$

Expanding the equation, we get:

$-3 = -3$

This statement is also true, which means that the point $(-2,-3)$ satisfies the second equation.

Conclusion

Since the point $(-2,-3)$ satisfies both equations, we can conclude that it is a solution to the system of equations.

Final Answer

The final answer is: A. Yes

Discussion

This problem requires the student to understand the concept of a solution to a system of linear equations. The student must be able to substitute the coordinates of a point into each equation and determine if the resulting statement is true. This problem also requires the student to understand the concept of a linear equation in slope-intercept form and how to simplify equations.

Tips and Tricks

• When substituting a point into an equation, make sure to use the correct coordinates.
• When simplifying an equation, make sure to follow the order of operations (PEMDAS).
• When determining if a point is a solution to a system of equations, make sure to check if the point satisfies both equations.

Related Problems

• Is the point $(1,2)$ a solution to the system of equations $\begin{array}{l} y = x + 1 \\ y = 2x - 1 \end{array}$?
• Is the point $(-1,-2)$ a solution to the system of equations $\begin{array}{l} y = -x + 1 \\ y = x - 2 \end{array}$?

Conclusion

In conclusion, the point $(-2,-3)$ is a solution to the system of equations $\begin{array}{l} y = 2x + 1 \\ y = x - 1 \end{array}$. This problem requires the student to understand the concept of a solution to a system of linear equations and how to substitute a point into each equation to determine if it is a solution.

Introduction

Systems of linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. In this article, we'll answer some common questions about systems of linear equations, including how to determine if a point is a solution, how to solve systems using substitution and elimination, and more.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are related to each other. Each equation is a linear equation, which means it can be written in the form 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 linear equations?

A: To determine if a point is a solution to a system of linear equations, you need to substitute the x and y coordinates of the point 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: How do I solve a system of linear equations using substitution?

A: To solve a system of linear equations using substitution, you need to solve one of the equations for one of the variables and then substitute that expression into the other equation. This will give you a new equation with one variable, which you can then solve.

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

A: To solve a system of linear equations using elimination, you need to add or subtract the equations in a way that eliminates one of the variables. This will give you a new equation with one variable, which you can then solve.

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

A: A system of linear equations consists of two or more linear equations, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that cannot be written in the form y = mx + b, where m is the slope and b is the y-intercept.

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

A: To graph a system of linear equations, you need to graph each equation separately and then find the point of intersection between the two lines. This point of intersection is the solution to the system.

Q: What is the point of intersection in a system of linear equations?

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

Q: Can a system of linear equations have more than one solution?

A: No, a system of linear equations can only have one solution. If a system has more than one solution, then it is inconsistent.

Q: Can a system of linear equations have no solution?

A: Yes, a system of linear equations can have no solution. This occurs when the two lines are parallel and never intersect.

Q: How do I determine if a system of linear equations is consistent or inconsistent?

A: To determine if a system of linear equations is consistent or inconsistent, you need to check if the two lines intersect. If the lines intersect, then the system is consistent. If the lines are parallel and never intersect, then the system is inconsistent.

Q: What is the difference between a consistent system and an inconsistent system?

A: A consistent system is a system of linear equations that has a solution, while an inconsistent system is a system of linear equations that has no solution.

Q: Can a system of linear equations have infinitely many solutions?

A: No, a system of linear equations can only have one solution or no solution. If a system has infinitely many solutions, then it is not a system of linear equations.

Conclusion

In conclusion, systems of linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. By following the steps outlined in this article, you can determine if a point is a solution to a system of linear equations, solve systems using substitution and elimination, and more.

Final Tips and Tricks

• When solving a system of linear equations, make sure to check if the two lines intersect.
• When graphing a system of linear equations, make sure to graph each equation separately.
• When determining if a system of linear equations is consistent or inconsistent, make sure to check if the two lines intersect.
• When solving a system of linear equations, make sure to follow the order of operations (PEMDAS).

Related Problems

• Solve the system of linear equations $\begin{array}{l} y = 2x + 1 \\ y = x - 1 \end{array}$ using substitution and elimination.
• Determine if the point $(1,2)$ is a solution to the system of linear equations $\begin{array}{l} y = x + 1 \\ y = 2x - 1 \end{array}$.
• Graph the system of linear equations $\begin{array}{l} y = x + 1 \\ y = 2x - 1 \end{array}$ and find the point of intersection.

Conclusion

In conclusion, systems of linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. By following the steps outlined in this article, you can determine if a point is a solution to a system of linear equations, solve systems using substitution and elimination, and more.