Which Of The Following Best Describes The Solution To The System Of Equations Below?${ \begin{array}{r} 4x + 6y = 10 \ 8x + 12y = 20 \end{array} }$

Mar 13, 2025 by ADMIN 149 views

**Solving Systems of Linear Equations: A Step-by-Step Guide**

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore the solution to a system of linear equations and provide a step-by-step guide on how to solve it.

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve the same variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. The system of equations can be represented graphically as a set of lines on a coordinate plane.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{r} 4x + 6y = 10 \\ 8x + 12y = 20 \end{array} \}

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the First Equation for x

We can solve the first equation for x by isolating x on one side of the equation.

4x + 6y = 10

Subtract 6y from both sides:

4x = 10 - 6y

Divide both sides by 4:

x = (10 - 6y) / 4

Step 2: Substitute the Expression for x into the Second Equation

Now that we have an expression for x, we can substitute it into the second equation.

8x + 12y = 20

Substitute x = (10 - 6y) / 4 into the equation:

8((10 - 6y) / 4) + 12y = 20

Simplify the equation:

2(10 - 6y) + 12y = 20

Expand and simplify:

20 - 12y + 12y = 20

Simplify further:

20 = 20

This is a true statement, which means that the system of equations has infinitely many solutions.

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.

Step 1: Multiply the First Equation by 2

To eliminate the variable x, we can multiply the first equation by 2.

4x + 6y = 10

Multiply both sides by 2:

8x + 12y = 20

Step 2: Subtract the Second Equation from the First Equation

Now that we have two equations with the same coefficients for x, we can subtract the second equation from the first equation to eliminate x.

8x + 12y = 20

Subtract 8x + 12y = 20 from 8x + 12y = 20:

0 = 0

This is a true statement, which means that the system of equations has infinitely many solutions.

Conclusion

In conclusion, the solution to the system of equations is that it has infinitely many solutions. This is because the two equations are linearly dependent, meaning that one equation is a multiple of the other. As a result, the system of equations has an infinite number of solutions.

Real-World Applications

Solving systems of linear equations has numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, systems of linear equations can be used to model the motion of objects under the influence of forces. In engineering, systems of linear equations can be used to design and optimize systems such as electrical circuits and mechanical systems.

Final Thoughts

Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to solve systems of linear equations, we can gain a deeper understanding of the underlying mathematical concepts and develop problem-solving skills that can be applied to a wide range of fields.

Key Takeaways

A system of linear equations is a set of two or more linear equations that involve the same variables.
The system of equations can be represented graphically as a set of lines on a coordinate plane.
The substitution method and elimination method are two common methods for solving systems of linear equations.
The system of equations has infinitely many solutions because the two equations are linearly dependent.

Recommended Resources

Khan Academy: Systems of Linear Equations
MIT OpenCourseWare: Linear Algebra
Wolfram MathWorld: Systems of Linear Equations

Practice Problems

Solve the system of equations: x + 2y = 3 and 2x + 4y = 6
Solve the system of equations: x - 2y = 1 and 2x - 4y = 2

Conclusion

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Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I know if a system of linear equations has a unique solution, infinitely many solutions, or no solution?

A: To determine the number of solutions to a system of linear equations, you can use the following criteria:

If the two equations are parallel (i.e., they have the same slope but different y-intercepts), the system has no solution.
If the two equations are identical, the system has infinitely many solutions.
If the two equations are not parallel and not identical, the system has a unique solution.

Q: What is the difference between the substitution method and the elimination method?

A: 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: Can I use the substitution method or the elimination method to solve a system of linear equations with more than two variables?

A: Yes, you can use either the substitution method or the elimination method to solve a system of linear equations with more than two variables. However, the elimination method may be more efficient for systems with more than two variables.

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

A: A system of linear equations is consistent if it has at least one solution. A system of linear equations is inconsistent if it has no solution.

Q: Can I use a graphing calculator to solve a system of linear equations?

A: Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can be used to graph the equations and find the point of intersection, which represents the solution to the system.

Q: How do I determine the number of solutions to a system of linear equations using a graphing calculator?

A: To determine the number of solutions to a system of linear equations using a graphing calculator, follow these steps:

Graph the two equations on the same coordinate plane.
If the two equations intersect at a single point, the system has a unique solution.
If the two equations are parallel, the system has no solution.
If the two equations are identical, the system has infinitely many solutions.

Q: Can I use a computer algebra system (CAS) to solve a system of linear equations?

A: Yes, you can use a computer algebra system (CAS) to solve a system of linear equations. CAS software can be used to solve systems of linear equations using various methods, including the substitution method and the elimination method.

Q: How do I determine the number of solutions to a system of linear equations using a CAS?

A: To determine the number of solutions to a system of linear equations using a CAS, follow these steps:

Enter the system of linear equations into the CAS software.
Use the CAS software to solve the system of linear equations.
The CAS software will display the solution to the system, which may be a single point, a line, or a plane, depending on the number of variables and the number of equations.

Q: Can I use a matrix to solve a system of linear equations?

A: Yes, you can use a matrix to solve a system of linear equations. A matrix can be used to represent the coefficients of the system of linear equations and the constants on the right-hand side of the equations.

Q: How do I use a matrix to solve a system of linear equations?

A: To use a matrix to solve a system of linear equations, follow these steps:

Represent the coefficients of the system of linear equations as a matrix.
Represent the constants on the right-hand side of the equations as a matrix.
Use the matrix to solve the system of linear equations.

Q: What is the advantage of using a matrix to solve a system of linear equations?

A: The advantage of using a matrix to solve a system of linear equations is that it can be used to solve systems of linear equations with more than two variables and more than two equations.

Q: Can I use a matrix to solve a system of linear equations with complex coefficients?

A: Yes, you can use a matrix to solve a system of linear equations with complex coefficients. However, the matrix must be a square matrix, and the determinant of the matrix must be non-zero.

Q: How do I determine the number of solutions to a system of linear equations using a matrix?

A: To determine the number of solutions to a system of linear equations using a matrix, follow these steps:

Represent the coefficients of the system of linear equations as a matrix.
Represent the constants on the right-hand side of the equations as a matrix.
Use the matrix to solve the system of linear equations.
If the matrix has a non-zero determinant, the system has a unique solution.
If the matrix has a zero determinant, the system has no solution.

Q: Can I use a matrix to solve a system of linear equations with more than two variables?

A: Yes, you can use a matrix to solve a system of linear equations with more than two variables. However, the matrix must be a square matrix, and the determinant of the matrix must be non-zero.

Q: How do I determine the number of solutions to a system of linear equations using a matrix with more than two variables?

A: To determine the number of solutions to a system of linear equations using a matrix with more than two variables, follow these steps:

Represent the coefficients of the system of linear equations as a matrix.
Represent the constants on the right-hand side of the equations as a matrix.
Use the matrix to solve the system of linear equations.
If the matrix has a non-zero determinant, the system has a unique solution.
If the matrix has a zero determinant, the system has no solution.

Q: Can I use a matrix to solve a system of linear equations with complex coefficients and more than two variables?

A: Yes, you can use a matrix to solve a system of linear equations with complex coefficients and more than two variables. However, the matrix must be a square matrix, and the determinant of the matrix must be non-zero.

Q: How do I determine the number of solutions to a system of linear equations using a matrix with complex coefficients and more than two variables?

A: To determine the number of solutions to a system of linear equations using a matrix with complex coefficients and more than two variables, follow these steps:

Represent the coefficients of the system of linear equations as a matrix.
Represent the constants on the right-hand side of the equations as a matrix.
Use the matrix to solve the system of linear equations.
If the matrix has a non-zero determinant, the system has a unique solution.
If the matrix has a zero determinant, the system has no solution.

Conclusion

In conclusion, solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to solve systems of linear equations, we can gain a deeper understanding of the underlying mathematical concepts and develop problem-solving skills that can be applied to a wide range of fields.

Key Takeaways

A system of linear equations is a set of two or more linear equations that involve the same variables.
The substitution method and the elimination method are two common methods for solving systems of linear equations.
A matrix can be used to represent the coefficients of the system of linear equations and the constants on the right-hand side of the equations.
The determinant of the matrix must be non-zero for the system to have a unique solution.