Determine If The Point { (-1, 6)$}$ Is A Solution To The System Of Equations: $ \begin{array}{l} 6x + 3y = 18 \ 2x + Y = 7 \end{array} }$2. Solve The System Of Equations $[ \begin{array {r} 2x +

Introduction

Systems of linear equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will explore how to determine if a given point is a solution to a system of linear equations and how to solve a system of linear equations using the substitution and elimination methods.

Determining if a Point is a Solution to a System of Equations

To determine if a point is a solution to a system of linear equations, we need to substitute the coordinates of the point into each equation and check if the resulting statement is true. Let's consider the following system of equations:

$${ \begin{array}{l} 6x + 3y = 18 \\ 2x + y = 7 \end{array} }$$

We are given the point ${(-1, 6)}$. To determine if this point is a solution to the system of equations, we need to substitute $x = -1$ and $y = 6$ into each equation.

Substituting the Point into the First Equation

Let's substitute $x = -1$ and $y = 6$ into the first equation:

$6x + 3y = 18$

$6(-1) + 3(6) = 18$

$-6 + 18 = 18$

$12 = 18$

The resulting statement is false, which means that the point ${(-1, 6)}$ is not a solution to the first equation.

Substituting the Point into the Second Equation

Let's substitute $x = -1$ and $y = 6$ into the second equation:

$2x + y = 7$

$2(-1) + 6 = 7$

$-2 + 6 = 7$

$4 = 7$

The resulting statement is false, which means that the point ${(-1, 6)}$ is not a solution to the second equation.

Conclusion

Since the point ${(-1, 6)}$ is not a solution to either equation, it is not a solution to the system of equations.

Solving a System of Linear Equations

Now that we have determined that the point ${(-1, 6)}$ is not a solution to the system of equations, let's solve the system of equations using the substitution method.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Let's solve the second equation for $y$:

$2x + y = 7$

$y = 7 - 2x$

Now, let's substitute this expression for $y$ into the first equation:

$6x + 3y = 18$

$6x + 3(7 - 2x) = 18$

$6x + 21 - 6x = 18$

$21 = 18$

The resulting statement is false, which means that the system of equations has no solution.

Elimination Method

The elimination method involves adding or subtracting the equations to eliminate one variable. Let's multiply the second equation by 3 to make the coefficients of $y$ in both equations equal:

$2x + y = 7$

$3(2x + y) = 3(7)$

$6x + 3y = 21$

Now, let's subtract the first equation from the second equation:

$(6x + 3y) - (6x + 3y) = 21 - 18$

$0 = 3$

The resulting statement is false, which means that the system of equations has no solution.

Conclusion

We have determined that the system of equations has no solution using both the substitution and elimination methods.

Discussion

Systems of linear equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we have explored how to determine if a given point is a solution to a system of linear equations and how to solve a system of linear equations using the substitution and elimination methods. We have also discussed the importance of checking the validity of the resulting statements to ensure that the solution is correct.

Conclusion

In conclusion, solving systems of linear equations is a crucial skill in mathematics, and it has numerous applications in various fields. By understanding how to determine if a given point is a solution to a system of linear equations and how to solve a system of linear equations using the substitution and elimination methods, we can apply this knowledge to real-world problems and make informed decisions.

References

• [1] "Systems of Linear Equations" by Math Open Reference
• [2] "Solving Systems of Linear Equations" by Khan Academy
• [3] "Systems of Linear Equations" by Wolfram MathWorld

Further Reading

• "Linear Algebra" by Gilbert Strang
• "Linear Algebra and Its Applications" by Gilbert Strang
• "Systems of Linear Equations" by MIT OpenCourseWare

Glossary

• System of linear equations: A set of two or more linear equations that are solved simultaneously.
• Substitution method: A method of solving a system of linear equations by substituting one equation into the other equation.
• Elimination method: A method of solving a system of linear equations by adding or subtracting the equations to eliminate one variable.
• Linear equation: An equation in which the highest power of the variable is 1.
• Variable: A symbol or expression that represents a value that can change.
  Frequently Asked Questions (FAQs) About Systems of Linear Equations ====================================================================

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 solved simultaneously. Each equation in the system is a linear equation, which means that the highest power of the variable is 1.

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 coordinates of the point into each equation and check if the resulting statement is true. If the resulting statement is true for both equations, then the point is a solution to the system of equations.

Q: What are the two main methods for solving systems of linear equations?

A: The two main methods for solving systems of linear equations are the substitution method and the elimination method.

• The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
• The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I use the substitution method to solve a system of linear equations?

A: To use the substitution method, you need to solve one equation for one variable and then substitute that expression into the other equation. For example, if you have the following system of equations:

$${ \begin{array}{l} 6x + 3y = 18 \\ 2x + y = 7 \end{array} }$$

You can solve the second equation for y:

$y = 7 - 2x$

Then, you can substitute this expression for y into the first equation:

$6x + 3(7 - 2x) = 18$

Q: How do I use the elimination method to solve a system of linear equations?

A: To use the elimination method, you need to add or subtract the equations to eliminate one variable. For example, if you have the following system of equations:

$${ \begin{array}{l} 6x + 3y = 18 \\ 2x + y = 7 \end{array} }$$

You can multiply the second equation by 3 to make the coefficients of y in both equations equal:

$6x + 3y = 18$

$6x + 3y = 21$

Then, you can subtract the first equation from the second equation:

$(6x + 3y) - (6x + 3y) = 21 - 18$

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

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. A system of nonlinear equations is a set of two or more nonlinear equations that are solved simultaneously. Nonlinear equations are equations in which the highest power of the variable is greater than 1.

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 equations are inconsistent, meaning that they cannot be true at the same time.

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

A: Yes, a system of linear equations can have an infinite number of solutions. This occurs when the equations are dependent, meaning that they are equivalent to each other.

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

A: To determine if a system of linear equations has a unique solution, no solution, or an infinite number of solutions, you need to examine the equations and determine if they are consistent, inconsistent, or dependent.

• If the equations are consistent and independent, then the system has a unique solution.
• If the equations are inconsistent, then the system has no solution.
• If the equations are dependent, then the system has an infinite number of solutions.

Conclusion

In conclusion, systems of linear equations are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding how to determine if a given point is a solution to a system of linear equations and how to solve a system of linear equations using the substitution and elimination methods, you can apply this knowledge to real-world problems and make informed decisions.

References

• [1] "Systems of Linear Equations" by Math Open Reference
• [2] "Solving Systems of Linear Equations" by Khan Academy
• [3] "Systems of Linear Equations" by Wolfram MathWorld

Further Reading

• "Linear Algebra" by Gilbert Strang
• "Linear Algebra and Its Applications" by Gilbert Strang
• "Systems of Linear Equations" by MIT OpenCourseWare

Glossary

• System of linear equations: A set of two or more linear equations that are solved simultaneously.
• Substitution method: A method of solving a system of linear equations by substituting one equation into the other equation.
• Elimination method: A method of solving a system of linear equations by adding or subtracting the equations to eliminate one variable.
• Linear equation: An equation in which the highest power of the variable is 1.
• Variable: A symbol or expression that represents a value that can change.