Plotting the equations on a coordinate plane

Oct 13, 2025 by windailys.com 67 views

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. 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 solution of the system.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{l} 7x + 7y = -49 \\ 7x - 2y = 5 \end{array} \}

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.

Method of Substitution

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

Let's start by solving the first equation for x:

7x + 7y = -49

Subtracting 7y from both sides gives:

7x = -49 - 7y

Dividing both sides by 7 gives:

x = (-49 - 7y) / 7

Now, let's substitute this expression for x into the second equation:

7x - 2y = 5

Substituting x = (-49 - 7y) / 7 into this equation gives:

7((-49 - 7y) / 7) - 2y = 5

Simplifying this equation gives:

-49 - 7y - 2y = 5

Combine like terms:

-49 - 9y = 5

Add 49 to both sides:

-9y = 54

Dividing both sides by -9 gives:

y = -54 / 9

y = -6

Now that we have found the value of y, we can substitute this value back into one of the original equations to find the value of x.

Let's substitute y = -6 into the first equation:

7x + 7y = -49

Substituting y = -6 into this equation gives:

7x + 7(-6) = -49

Simplifying this equation gives:

7x - 42 = -49

Add 42 to both sides:

7x = -7

Dividing both sides by 7 gives:

x = -7 / 7

x = -1

Method of Elimination

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

Let's start by adding the two equations together:

7x + 7y = -49

7x - 2y = 5

Adding these two equations gives:

14x + 5y = -44

Now, let's subtract the second equation from the first equation:

7x + 7y = -49

7x - 2y = 5

Subtracting these two equations gives:

9y = -54

Dividing both sides by 9 gives:

y = -54 / 9

y = -6

Now that we have found the value of y, we can substitute this value back into one of the original equations to find the value of x.

Let's substitute y = -6 into the first equation:

7x + 7y = -49

Substituting y = -6 into this equation gives:

7x + 7(-6) = -49

Simplifying this equation gives:

7x - 42 = -49

Add 42 to both sides:

7x = -7

Dividing both sides by 7 gives:

x = -7 / 7

x = -1

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 that the solution to the system is x = -1 and y = -6.

Final Answer

The final answer is $\boxed{(-1, -6)}$ .

Tips and Tricks

When solving a system of linear equations, it's essential to check your work by plugging the solution back into the original equations.
The method of substitution and elimination can be used to solve systems of linear equations with two or more variables.
When using the method of elimination, make sure to add or subtract the equations in a way that eliminates one of the variables.
When using the method of substitution, make sure to substitute the expression for one variable into the other equation correctly.

Common Mistakes

One common mistake when solving a system of linear equations is to forget to check your work by plugging the solution back into the original equations.
Another common mistake is to add or subtract the equations incorrectly when using the method of elimination.
When using the method of substitution, make sure to substitute the expression for one variable into the other equation correctly.

Real-World Applications

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

Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Computer Science: Systems of linear equations are used in computer science to solve problems, such as finding the shortest path in a graph and solving systems of linear equations.
Economics: Systems of linear equations are used in economics to model real-world problems, such as the behavior of supply and demand and the behavior of economic systems.

Conclusion

In conclusion, solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. By using the method of substitution and elimination, we can solve systems of linear equations with two or more variables. It's essential to check your work by plugging the solution back into the original equations and to avoid common mistakes, such as adding or subtracting the equations incorrectly.

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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 are solved simultaneously to find the values of the variables.

Q: How do I know which method to use to solve a system of linear equations?

A: The choice of method depends on the type of system and the variables involved. If the system has two variables and the equations are in the form of ax + by = c, you can use the method of substitution or elimination. If the system has more than two variables or the equations are not in the form of ax + by = c, you may need to use other methods, such as graphing or matrices.

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 adding or subtracting the equations to eliminate one of the variables.

Q: How do I check my work when solving a system of linear equations?

A: To check your work, plug the solution back into the original equations and make sure that both equations are satisfied. If the solution does not satisfy both equations, you may need to re-evaluate your work.

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

A: Some common mistakes to avoid include:

Forgetting to check your work by plugging the solution back into the original equations.
Adding or subtracting the equations incorrectly when using the method of elimination.
Substituting the expression for one variable into the other equation incorrectly when using the method of substitution.

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

A: To use matrices to solve a system of linear equations, you can represent the system as an augmented matrix and then use row operations to transform the matrix into row-echelon form. From there, you can use back-substitution to find the solution.

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

A: Some real-world applications of solving systems of linear equations include:

Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Computer Science: Systems of linear equations are used in computer science to solve problems, such as finding the shortest path in a graph and solving systems of linear equations.
Economics: Systems of linear equations are used in economics to model real-world problems, such as the behavior of supply and demand and the behavior of economic systems.

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

A: To graph a system of linear equations, you can plot the equations on a coordinate plane and then find the point of intersection. The point of intersection represents the solution to the system.

Q: What is the point of intersection in a system of linear equations?

A: The point of intersection in a system of linear equations is the point where the two lines intersect. This point represents the solution to the system.

Q: How do I find the point of intersection in a system of linear equations?

A: To find the point of intersection in a system of linear equations, you can plot the equations on a coordinate plane and then find the point where the two lines intersect. You can also use algebraic methods, such as substitution or elimination, to find the solution.

Q: What are some tips for solving systems of linear equations?

A: Some tips for solving systems of linear equations include:

Make sure to check your work by plugging the solution back into the original equations.
Use the method of substitution or elimination to solve the system.
Avoid common mistakes, such as adding or subtracting the equations incorrectly.
Use matrices to solve the system if it is in the form of an augmented matrix.

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 lines intersect at a single point. If the lines are parallel, the system has no solution. If the lines are coincident, the system has infinitely many solutions.

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

A: A system of linear equations is a set of linear equations that are solved simultaneously to find the values of the variables. A system of nonlinear equations is a set of nonlinear equations that are solved simultaneously to find the values of the variables. Nonlinear equations are equations that are not in the form of ax + by = c.

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

A: To solve a system of nonlinear equations, you can use algebraic methods, such as substitution or elimination, or numerical methods, such as the Newton-Raphson method. You can also use graphical methods, such as plotting the equations on a coordinate plane and finding the point of intersection.

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

A: Some real-world applications of solving systems of nonlinear equations include:

Physics and Engineering: Systems of nonlinear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Computer Science: Systems of nonlinear equations are used in computer science to solve problems, such as finding the shortest path in a graph and solving systems of nonlinear equations.
Economics: Systems of nonlinear equations are used in economics to model real-world problems, such as the behavior of supply and demand and the behavior of economic systems.

Q: How do I use numerical methods to solve a system of nonlinear equations?

A: To use numerical methods to solve a system of nonlinear equations, you can use the Newton-Raphson method or other iterative methods. These methods involve making an initial guess for the solution and then iteratively improving the guess until the solution is found.

Q: What are some common numerical methods for solving systems of nonlinear equations?

A: Some common numerical methods for solving systems of nonlinear equations include:

Newton-Raphson method: This method involves making an initial guess for the solution and then iteratively improving the guess until the solution is found.
Bisection method: This method involves finding the solution by repeatedly dividing the interval in which the solution lies.
Secant method: This method involves finding the solution by repeatedly using the secant line to approximate the solution.

Q: How do I use graphical methods to solve a system of nonlinear equations?

A: To use graphical methods to solve a system of nonlinear equations, you can plot the equations on a coordinate plane and then find the point of intersection. The point of intersection represents the solution to the system.

Q: What are some common graphical methods for solving systems of nonlinear equations?

A: Some common graphical methods for solving systems of nonlinear equations include:

Plotting the equations on a coordinate plane: This method involves plotting the equations on a coordinate plane and then finding the point of intersection.
Using a graphing calculator: This method involves using a graphing calculator to plot the equations and find the point of intersection.
Using a computer program: This method involves using a computer program to plot the equations and find the point of intersection.