Solve The Following System Of Equations.${ \begin{array}{l} -8x + 3y = 7 \ 13x - 3y = -17 \end{array} }${$ X = \square $}${$ Y = \square $}$

Mar 10, 2025 by ADMIN 144 views

**Solving a System of Linear Equations**

Introduction

Solving a system 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 focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the values of x and y.

The System of Equations

The given system of equations is:

{ \begin{array}{l} -8x + 3y = 7 \\ 13x - 3y = -17 \end{array} \}

Our goal is to find the values of x and y that satisfy both equations.

Method of Substitution

One way to solve this system of equations is by using the method of substitution. We can solve one equation for one variable and then substitute that expression into the other equation.

Let's solve the first equation for y:

{ -8x + 3y = 7 \}

{ 3y = 7 + 8x \}

{ y = \frac{7 + 8x}{3} \}

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

{ 13x - 3y = -17 \}

{ 13x - 3\left(\frac{7 + 8x}{3}\right) = -17 \}

{ 13x - \frac{7 + 8x}{1} = -17 \}

{ 13x - 7 - 8x = -17 \}

{ 5x - 7 = -17 \}

{ 5x = -10 \}

{ x = -2 \}

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

{ -8x + 3y = 7 \}

{ -8(-2) + 3y = 7 \}

{ 16 + 3y = 7 \}

{ 3y = -9 \}

{ y = -3 \}

Method of Elimination

Another way to solve this system of equations is by using the method of elimination. We can multiply both equations by necessary multiples such that the coefficients of y's in both equations are the same:

{ \begin{array}{l} -8x + 3y = 7 \\ 13x - 3y = -17 \end{array} \}

Multiply the first equation by 1 and the second equation by 1:

{ \begin{array}{l} -8x + 3y = 7 \\ 13x - 3y = -17 \end{array} \}

Add both equations to eliminate the y variable:

{ -8x + 13x + 3y - 3y = 7 - 17 \}

{ 5x = -10 \}

{ x = -2 \}

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

{ -8x + 3y = 7 \}

{ -8(-2) + 3y = 7 \}

{ 16 + 3y = 7 \}

{ 3y = -9 \}

{ y = -3 \}

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have found the values of x and y that satisfy both equations. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same, and then adding both equations to eliminate the y variable.

Final Answer

The final answer is:

{ x = \boxed{-2} \}

{ y = \boxed{-3} \}

Discussion

This system of equations can be solved using other methods such as the method of matrices or the method of graphs. However, the method of substitution and elimination are the most commonly used methods in solving systems of linear equations.

Real-World Applications

Solving systems of linear equations has many real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, the motion of an object can be described by a system of linear equations. In engineering, the design of a bridge can be described by a system of linear equations. In economics, the supply and demand of a product can be described by a system of linear equations. In computer science, the solution of a system of linear equations can be used to solve problems such as image processing and computer vision.

Future Work

In the future, we can explore other methods of solving systems of linear equations such as the method of matrices or the method of graphs. We can also apply the method of substitution and elimination to solve systems of linear equations with more than two variables.

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra and Its Applications" by David C. Lay

Note: The references provided are just examples and are not actual references used in this article.

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we discussed the method of substitution and elimination to solve a system of two linear equations with two variables. In this article, we will provide a Q&A section to help you better understand the concept and solve similar problems.

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 two or more variables. Each equation is a statement that two expressions are equal, and the variables are the unknown values that we need to find.

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

A: A system of linear equations has a solution if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.

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

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same, and then adding both equations to eliminate the y variable.

Q: How do I choose between the method of substitution and the method of elimination?

A: You can choose between the two methods based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, then the method of elimination is easier to use. If the coefficients of one variable are different in both equations, then the method of substitution is easier to use.

Q: What if I have a system of linear equations with more than two variables?

A: If you have a system of linear equations with more than two variables, then you can use the method of substitution or the method of elimination to solve for one variable, and then substitute that expression into the other equations to solve for the remaining variables.

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

A: Yes, you can use a calculator to solve a system of linear equations. Most calculators have a built-in function to solve systems of linear equations, and you can enter the coefficients of the variables and the constants to find the solution.

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

A: If you have a system of linear equations with no solution, then the two equations are inconsistent, meaning that they contradict each other. In this case, you can use the method of substitution or the method of elimination to find the values of the variables that make one equation true, but not the other.

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

A: Yes, you can use a graph to solve a system of linear equations. If the two equations are linear, then you can graph the two lines on a coordinate plane and find the point of intersection, which is the solution to the system.

Q: What are some real-world applications of solving systems of linear equations?

A: Solving systems of linear equations has many real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, the motion of an object can be described by a system of linear equations. In engineering, the design of a bridge can be described by a system of linear equations. In economics, the supply and demand of a product can be described by a system of linear equations. In computer science, the solution of a system of linear equations can be used to solve problems such as image processing and computer vision.