Solve The System Of Equations.$\[ \begin{array}{l} y = 6x + 13 \\ y = 2x + 1 \end{array} \\]a. \[$(5, 3)\$\] B. \[$(-3, -5)\$\] C. \[$(3, -5)\$\] D. No Solution Please Select The Best Answer From The Choices

Feb 27, 2025 by ADMIN 212 views

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

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the solution to a set of two or more linear equations that are related to each other. In this article, we will focus on solving a system of two linear equations in two variables. We will use the method of substitution and elimination to find the solution.

What are Systems of Linear Equations?

A system of linear equations is a set of two or more linear equations that are related to each other. 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.

Example: Solving a System of Two Linear Equations

Let's consider the following system of two linear equations:

y = 6x + 13 y = 2x + 1

We are given four possible solutions to this system of equations:

a. (5, 3) b. (-3, -5) c. (3, -5) d. No solution

To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.

Method of Substitution

The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. In this case, we can solve the first equation for y:

y = 6x + 13

Now, we can substitute this expression for y into the second equation:

6x + 13 = 2x + 1

Simplifying the Equation

To simplify the equation, we can subtract 2x from both sides:

4x + 13 = 1

Next, we can subtract 13 from both sides:

4x = -12

Solving for x

To solve for x, we can divide both sides by 4:

x = -12/4 x = -3

Finding the Value of y

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:

y = 6x + 13 y = 6(-3) + 13 y = -18 + 13 y = -5

Checking the Solution

To check the solution, we can substitute the values of x and y into both original equations:

y = 6x + 13 -5 = 6(-3) + 13 -5 = -18 + 13 -5 = -5

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

The solution (x, y) = (-3, -5) satisfies both original equations.

Conclusion

In this article, we have solved a system of two linear equations using the method of substitution. We have found the solution (x, y) = (-3, -5) and checked it to ensure that it satisfies both original equations. This is just one example of how to solve a system of linear equations. There are many other methods and techniques that can be used to solve systems of linear equations, including the method of elimination and the use of matrices.

Final Answer

The final answer is:

b. (-3, -5)

Discussion

This problem is a classic example of a system of linear equations. It involves finding the solution to a set of two linear equations that are related to each other. The method of substitution is a powerful tool for solving systems of linear equations, and it can be used to solve a wide range of problems. In this case, we have used the method of substitution to find the solution (x, y) = (-3, -5). This solution satisfies both original equations, and it is the only solution to the system of equations.

Additional Resources

For more information on solving systems of linear equations, please see the following resources:

Khan Academy: Systems of Linear Equations
Mathway: Systems of Linear Equations
Wolfram Alpha: Systems of Linear Equations

References

Hall, R. (2013). Algebra: A Comprehensive Introduction. McGraw-Hill Education.
Larson, R. (2014). Algebra and Trigonometry. Cengage Learning.
Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
Solving Systems of Linear Equations: Q&A =============================================

Introduction

In our previous article, we discussed how to solve systems of linear equations using the method of substitution. We also provided a step-by-step guide on how to solve a system of two linear equations. In this article, we will answer some frequently asked questions (FAQs) about solving systems of linear equations.

Q: What is a system of linear equations?

Q: How do I know if a system of linear equations has a solution?

A system of linear equations has a solution if the two lines intersect at a single point. If the lines are parallel, then the system has no solution. If the lines are the same, then the system has infinitely many solutions.

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, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that are not in the form of ax + by = c.

Q: Can I use the method of substitution to solve a system of three or more linear equations?

Yes, you can use the method of substitution to solve a system of three or more linear equations. However, it may be more complicated and time-consuming.

Q: What is the method of elimination?

The method of elimination involves adding or subtracting the two equations to eliminate one of the variables. This method is often used when the coefficients of the variables are the same.

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

Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can plot the lines and find the intersection point.

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

A system of linear equations is a set of two or more linear equations, while a matrix equation is a set of linear equations represented in matrix form.

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

Yes, you can use matrices to solve a system of linear equations. Matrices can be used to represent the coefficients of the variables and the constants.

Q: What is the formula for solving a system of linear equations using matrices?

The formula for solving a system of linear equations using matrices is:

AX = B

where A is the matrix of coefficients, X is the matrix of variables, and B is the matrix of constants.

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

Yes, you can use a computer program to solve a system of linear equations. Computer programs such as MATLAB and Python can be used to solve systems of linear equations.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the method of substitution, the method of elimination, and the use of matrices to solve systems of linear equations. We have also provided information on how to use graphing calculators and computer programs to solve systems of linear equations.