Solve The System Of Equations:$\[ \begin{aligned} x - Y &= -10 \\ -x + 6y &= 25 \end{aligned} \\]

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.

The System of Equations

The given system of equations is:

$$\begin{aligned} x - y &= -10 \\ -x + 6y &= 25 \end{aligned}$$

Step 1: Write Down the System of Equations

The first step in solving a system of linear equations is to write down the system of equations. In this case, we have two equations with two variables, x and y.

Step 2: Multiply the First Equation by 6

To eliminate the variable y, we can multiply the first equation by 6. This will give us:

$$6(x - y) = 6(-10)$$

Expanding the left-hand side, we get:

$$6x - 6y = -60$$

Step 3: Add the Two Equations

Now, we can add the two equations to eliminate the variable y. Adding the two equations, we get:

$$(6x - 6y) + (-x + 6y) = -60 + 25$$

Simplifying the left-hand side, we get:

$$5x = -35$$

Step 4: Solve for x

Now that we have eliminated the variable y, we can solve for x. Dividing both sides by 5, we get:

$$x = -7$$

Step 5: Substitute x into One of the Original Equations

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 substitute x into the first equation:

$$x - y = -10$$

Substituting x = -7, we get:

$$-7 - y = -10$$

Step 6: Solve for y

Now, we can solve for y. Adding 7 to both sides, we get:

$$-y = -3$$

Multiplying both sides by -1, we get:

$$y = 3$$

Conclusion

In this article, we have solved a system of two linear equations with two variables. We used the given system of equations as an example and provided a step-by-step guide on how to solve it. We multiplied the first equation by 6, added the two equations, solved for x, and substituted x into one of the original equations to find the value of y. The final solution is x = -7 and y = 3.

Tips and Tricks

• When solving a system of linear equations, it's essential to eliminate one of the variables by multiplying one of the equations by a suitable constant.
• When adding the two equations, make sure to combine like terms.
• When substituting x into one of the original equations, make sure to use the correct value of x.

Real-World Applications

Solving systems of linear equations has numerous real-world applications, including:

• Physics: Solving systems of linear equations is essential in physics to describe the motion of objects.
• Engineering: Solving systems of linear equations is crucial in engineering to design and optimize systems.
• Economics: Solving systems of linear equations is used in economics to model and analyze economic systems.

Conclusion

Introduction

In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables. 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?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear equation, which means it can be written in the form 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 solution?

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 a system of linear equations and a system of nonlinear equations?

A system of linear equations involves linear equations, which means that each equation can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables. A system of nonlinear equations, on the other hand, involves nonlinear equations, which means that each equation cannot be written in the form ax + by = c.

Q: How do I solve a system of linear equations with three variables?

To solve a system of linear equations with three variables, you can use the same steps as solving a system of linear equations with two variables. However, you will need to eliminate one of the variables by multiplying one of the equations by a suitable constant, and then substitute the value of the eliminated variable into one of the original equations.

Q: What is the method of substitution?

The method of substitution is a technique used to solve a system of linear equations. It involves substituting the value of one variable into one of the original equations, and then solving for the other variable.

Q: What is the method of elimination?

The method of elimination is a technique used to solve a system of linear equations. It involves eliminating one of the variables by multiplying one of the equations by a suitable constant, and then adding the two equations to eliminate the variable.

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 be used to graph the two equations and find the point of intersection, which represents the solution to the system.

Q: What is the importance of solving systems of linear equations?

Solving systems of linear equations is an essential skill in mathematics and has numerous real-world applications. It is used in physics, engineering, economics, and many other fields to model and analyze complex systems.

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, Python, and R can be used to solve systems of linear equations and provide the solution.

Conclusion

In conclusion, solving systems of linear equations is a fundamental concept in mathematics, and it has numerous real-world applications. By understanding the basics of solving systems of linear equations, you can apply this skill to a wide range of problems in physics, engineering, economics, and many other fields.

Tips and Tricks

• When solving a system of linear equations, it's essential to eliminate one of the variables by multiplying one of the equations by a suitable constant.
• When adding the two equations, make sure to combine like terms.
• When substituting x into one of the original equations, make sure to use the correct value of x.

Real-World Applications

Solving systems of linear equations has numerous real-world applications, including:

• Physics: Solving systems of linear equations is essential in physics to describe the motion of objects.
• Engineering: Solving systems of linear equations is crucial in engineering to design and optimize systems.
• Economics: Solving systems of linear equations is used in economics to model and analyze economic systems.

Conclusion

In conclusion, solving systems of linear equations is a fundamental concept in mathematics, and it has numerous real-world applications. By understanding the basics of solving systems of linear equations, you can apply this skill to a wide range of problems in physics, engineering, economics, and many other fields.