Solve The Following System Of Equations:${ \begin{array}{l} -6x - 5y = 5 \ 4x - 7y = 7 \end{array} }$

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

What is a System of Linear Equations?

A system of linear equations is a set of equations that 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 linear equations can have one or more equations, and the variables can be either one or more.

Types of Systems of Linear Equations

There are three types of systems of linear equations:

  1. Consistent System: A consistent system of linear equations is one that has a solution. In other words, there is at least one set of values for the variables that satisfies all the equations in the system.
  2. Inconsistent System: An inconsistent system of linear equations is one that has no solution. In other words, there is no set of values for the variables that satisfies all the equations in the system.
  3. Dependent System: A dependent system of linear equations is one that has an infinite number of solutions. In other words, there are an infinite number of sets of values for the variables that satisfy all the equations in the system.

Solving a System of Linear Equations

There are several methods for solving a system of linear equations, including:

  1. Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  2. Elimination Method: This method involves adding or subtracting the equations in the system to eliminate one variable.
  3. Graphical Method: This method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.

Solving the Given System of Equations

The given system of equations is:

-6x - 5y = 5 4x - 7y = 7

We will use the elimination method to solve this system of equations.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one variable, we need to multiply the equations by necessary multiples such that the coefficients of one variable are the same.

Let's multiply the first equation by 7 and the second equation by 5:

-42x - 35y = 35 20x - 35y = 35

Step 2: Subtract the Second Equation from the First Equation

Now, let's subtract the second equation from the first equation to eliminate the variable y:

(-42x - 35y) - (20x - 35y) = 35 - 35 -62x = 0

Step 3: Solve for x

Now, let's solve for x:

-62x = 0 x = 0/(-62) x = 0

Step 4: Substitute x into One of the Original Equations

Now, let's substitute x into one of the original equations to solve for y:

-6x - 5y = 5 -6(0) - 5y = 5 -5y = 5 y = -1

Step 5: Write the Solution

The solution to the system of equations is x = 0 and y = -1.

Conclusion

In this article, we solved a system of two linear equations with two variables using the elimination method. We first multiplied the equations by necessary multiples to eliminate one variable, then subtracted the second equation from the first equation to eliminate the variable y. We then solved for x and substituted x into one of the original equations to solve for y. The solution to the system of equations is x = 0 and y = -1.

Example Use Cases

Solving a system of linear equations has many practical applications in real-life situations, such as:

  1. Physics: Solving a system of linear equations can help us find the position and velocity of an object in a two-dimensional space.
  2. Engineering: Solving a system of linear equations can help us find the stress and strain on a material in a two-dimensional space.
  3. Economics: Solving a system of linear equations can help us find the equilibrium price and quantity of a good in a two-dimensional space.

Tips and Tricks

Here are some tips and tricks for solving a system of linear equations:

  1. Use the elimination method: The elimination method is a powerful tool for solving a system of linear equations.
  2. Check your work: Always check your work to make sure that the solution satisfies all the equations in the system.
  3. Use a graphing calculator: A graphing calculator can help you visualize the equations in the system and find the point of intersection.

Conclusion

Introduction

In our previous article, we solved a system of two linear equations with two variables using the elimination method. In this article, we will answer some frequently asked questions about solving a system 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 involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.

Q: What are the different types of systems of linear equations?

A: There are three types of systems of linear equations:

  1. Consistent System: A consistent system of linear equations is one that has a solution. In other words, there is at least one set of values for the variables that satisfies all the equations in the system.
  2. Inconsistent System: An inconsistent system of linear equations is one that has no solution. In other words, there is no set of values for the variables that satisfies all the equations in the system.
  3. Dependent System: A dependent system of linear equations is one that has an infinite number of solutions. In other words, there are an infinite number of sets of values for the variables that satisfy all the equations in the system.

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

A: There are several methods for solving a system of linear equations, including:

  1. Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  2. Elimination Method: This method involves adding or subtracting the equations in the system to eliminate one variable.
  3. Graphical Method: This method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.

Q: What is the elimination method?

A: The elimination method is a method for solving a system of linear equations by adding or subtracting the equations in the system to eliminate one variable.

Q: How do I use the elimination method?

A: To use the elimination method, follow these steps:

  1. Multiply the equations by necessary multiples such that the coefficients of one variable are the same.
  2. Subtract the second equation from the first equation to eliminate the variable.
  3. Solve for the remaining variable.
  4. Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

Q: What are some common mistakes to avoid when solving a system of linear equations?

A: Some common mistakes to avoid when solving a system of linear equations include:

  1. Not checking your work: Always check your work to make sure that the solution satisfies all the equations in the system.
  2. Not using the correct method: Make sure to use the correct method for solving the system of linear equations.
  3. Not following the steps: Make sure to follow the steps for the elimination method carefully.

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 equations are consistent. In other words, there is at least one set of values for the variables that satisfies all the equations in the system.

Q: How do I know if a system of linear equations has no solution?

A: A system of linear equations has no solution if the equations are inconsistent. In other words, there is no set of values for the variables that satisfies all the equations in the system.

Q: How do I know if a system of linear equations has an infinite number of solutions?

A: A system of linear equations has an infinite number of solutions if the equations are dependent. In other words, there are an infinite number of sets of values for the variables that satisfy all the equations in the system.

Conclusion

Solving a system of linear equations is an important skill in mathematics and has many practical applications in real-life situations. In this article, we answered some frequently asked questions about solving a system of linear equations. We hope that this article has been helpful in clarifying any confusion you may have had about solving a system of linear equations.