Is ( 2 , 4 (2,4 ( 2 , 4 ] A Solution To This System Of Equations?${ \begin{array}{l} y = 6x - 8 \ y = 3x + 2 \end{array} }$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 presented with a system of two linear equations and asked to verify if the point $(2,4)$ is a solution. To do this, we'll substitute the coordinates of the point into each equation and check if the resulting statement is true.

Understanding the System of Equations

The given system consists of two linear equations:

1. $y = 6x - 8$
2. $y = 3x + 2$

These equations are in the slope-intercept form, where the slope-intercept form of a linear equation is given by $y = mx + b$, with $m$ representing the slope and $b$ representing the y-intercept.

Substituting the Point into Each Equation

To determine if the point $(2,4)$ is a solution, we'll substitute the x-coordinate and y-coordinate into each equation.

Substituting into Equation 1

For the first equation, $y = 6x - 8$, we'll substitute $x = 2$ and $y = 4$:

$4 = 6(2) - 8$

Expanding the equation, we get:

$4 = 12 - 8$

Simplifying further, we have:

$4 = 4$

This statement is true, indicating that the point $(2,4)$ satisfies the first equation.

Substituting into Equation 2

For the second equation, $y = 3x + 2$, we'll substitute $x = 2$ and $y = 4$:

$4 = 3(2) + 2$

Expanding the equation, we get:

$4 = 6 + 2$

Simplifying further, we have:

$4 = 8$

This statement is false, indicating that the point $(2,4)$ does not satisfy the second equation.

Conclusion

Based on the results of substituting the point $(2,4)$ into each equation, we can conclude that the point is a solution to the first equation but not the second equation. Since a solution to a system of equations must satisfy both equations, the point $(2,4)$ is not a solution to the given system.

The final answer is: B. No

Discussion

In this discussion, we've explored the process of determining whether a given point satisfies a system of linear equations. By substituting the coordinates of the point into each equation, we can verify if the point is a solution. This process is essential in various applications, such as solving systems of equations, graphing linear equations, and analyzing the behavior of linear functions.

Applications

The concept of determining whether a point is a solution to a system of equations has numerous applications in various fields, including:

• Linear Algebra: In linear algebra, systems of equations are used to represent linear transformations and solve for unknown variables.
• Graphing: Graphing linear equations involves determining the points that satisfy the equation, which is essential in understanding the behavior of linear functions.
• Optimization: In optimization problems, systems of equations are used to find the optimal solution, which may involve determining whether a given point satisfies the system.

Conclusion

In conclusion, determining whether a point is a solution to a system of equations is a crucial concept in mathematics. By substituting the coordinates of the point into each equation, we can verify if the point satisfies the system. This process has numerous applications in various fields, including linear algebra, graphing, and optimization.

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Introduction

In our previous article, we explored whether the point $(2,4)$ is a solution to the system of equations:

1. $y = 6x - 8$
2. $y = 3x + 2$

We concluded that the point is a solution to the first equation but not the second equation. In this Q&A article, we'll address some common questions related to this topic.

Q: What is a solution to a system of equations?

A: A solution to a system of equations is a point that satisfies all the equations in the system. In other words, it's a point that makes each equation true.

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

A: To determine if a point is a solution, substitute the coordinates of the point into each equation and check if the resulting statement is true. If the point satisfies all the equations, it's a solution.

Q: What if the point satisfies one equation but not the other?

A: If the point satisfies one equation but not the other, it's not a solution to the system. A solution must satisfy all the equations in the system.

Q: Can a point be a solution to a system of equations if it's not on the graph of either equation?

A: Yes, a point can be a solution to a system of equations even if it's not on the graph of either equation. This is because the point may satisfy the equations algebraically, even if it's not visible on the graph.

Q: How do I know if a point is on the graph of an equation?

A: To determine if a point is on the graph of an equation, substitute the coordinates of the point into the equation and check if the resulting statement is true. If the point satisfies the equation, it's on the graph.

Q: Can a system of equations have multiple solutions?

A: Yes, a system of equations can have multiple solutions. This occurs when the equations intersect at multiple points, and each point satisfies both equations.

Q: How do I find the solutions to a system of equations?

A: To find the solutions to a system of equations, you can use various methods, such as:

• Substitution method: Substitute one equation into the other to solve for the variables.
• Elimination method: Add or subtract the equations to eliminate one variable and solve for the other.
• Graphing method: Graph the equations on a coordinate plane and find the intersection points.

Q: What if I have a system of equations with no solutions?

A: If a system of equations has no solutions, it means that the equations are inconsistent, and there is no point that satisfies both equations.

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

A: Yes, a system of equations can have infinitely many solutions. This occurs when the equations are dependent, and one equation is a multiple of the other.

Conclusion

In this Q&A article, we've addressed some common questions related to determining whether a point is a solution to a system of equations. We've covered topics such as what a solution is, how to determine if a point is a solution, and how to find the solutions to a system of equations.