Use substitution and elimination methods

Oct 13, 2025 by windailys.com 144 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 given system of equations to demonstrate the steps involved in solving such a system.

The System of Equations

The given system of equations is:

${-5y - 20 = x}$
${0 = 24x - 15y + 75}$

Step 1: Simplify the First Equation

To simplify the first equation, we can isolate the variable x by adding 20 to both sides of the equation.

${-5y - 20 + 20 = x + 20}$

This simplifies to:

${-5y = x + 20}$

Step 2: Simplify the Second Equation

To simplify the second equation, we can isolate the variable x by adding 15y to both sides of the equation.

${0 + 15y = 24x + 75 + 15y}$

This simplifies to:

${15y = 24x + 75}$

Step 3: Solve for x in the First Equation

We can solve for x in the first equation by isolating x on one side of the equation.

${-5y = x + 20}$

Subtracting 20 from both sides gives:

${-5y - 20 = x}$

Step 4: Substitute the Expression for x into the Second Equation

We can substitute the expression for x from the first equation into the second equation.

${15y = 24(-5y - 20) + 75}$

Expanding the right-hand side gives:

${15y = -120y - 480 + 75}$

Step 5: Combine Like Terms

We can combine like terms on the right-hand side of the equation.

${15y = -120y - 405}$

Step 6: Add 120y to Both Sides

We can add 120y to both sides of the equation to get:

${135y = -405}$

Step 7: Divide Both Sides by 135

We can divide both sides of the equation by 135 to get:

${y = -\frac{405}{135}}$

This simplifies to:

${y = -3}$

Step 8: Substitute the Value of y into the First Equation

We can substitute the value of y into the first equation to solve for x.

${-5(-3) - 20 = x}$

Expanding the left-hand side gives:

${15 - 20 = x}$

This simplifies to:

${-5 = x}$

Conclusion

In this article, we have demonstrated the steps involved in solving a system of two linear equations with two variables. We have used the given system of equations to show how to simplify the equations, substitute expressions, and solve for the variables. The final solution to the system of equations is x = -5 and y = -3.

Applications of Solving Systems of Equations

Solving systems of equations has numerous applications in various fields, including:

Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
Computer Science: Systems of equations are used in computer graphics, game development, and machine learning.
Economics: Systems of equations are used to model economic systems, including supply and demand, inflation, and unemployment.
Biology: Systems of equations are used to model population dynamics, epidemiology, and ecology.

Tips for Solving Systems of Equations

Here are some tips for solving systems of equations:

Use substitution and elimination methods: These methods are useful for solving systems of equations with two variables.
Graph the equations: Graphing the equations can help you visualize the solution and identify any inconsistencies.
Check your work: Always check your work by plugging the solution back into the original equations.
Use technology: Technology, such as graphing calculators and computer software, can help you solve systems of equations quickly and accurately.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving systems of equations:

Not checking your work: Failing to check your work can lead to incorrect solutions.
Not using the correct method: Using the wrong method can lead to incorrect solutions.
Not considering all possible solutions: Failing to consider all possible solutions can lead to incorrect conclusions.
Not using technology: Failing to use technology can lead to errors and inconsistencies.

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Introduction

In our previous article, we demonstrated the steps involved in solving a system of two linear equations with two variables. In this article, we will answer some frequently asked questions about solving systems of equations.

Q: What is a system of linear equations?

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 if a system of equations has a solution?

A system of equations has a solution if the lines intersect at a single point. If the lines are parallel, the system has no solution. If the lines coincide, 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 linear equations, while a system of nonlinear equations consists of nonlinear equations. Nonlinear equations are equations that are not linear, such as quadratic or polynomial equations.

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

Solving a system of nonlinear equations can be more challenging than solving a system of linear equations. There are several methods for solving systems of nonlinear equations, including:

Graphical methods: Graphing the equations can help you visualize the solution and identify any inconsistencies.
Numerical methods: Numerical methods, such as the Newton-Raphson method, can be used to approximate the solution.
Algebraic methods: Algebraic methods, such as substitution and elimination, can be used to solve systems of nonlinear equations.

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

A system of equations consists of equations, while a system of inequalities consists of inequalities. Inequalities are statements that compare two expressions, such as "x > 2" or "y < 3".

Q: How do I solve a system of inequalities?

Solving a system of inequalities involves finding the region that satisfies all the inequalities. This can be done using graphical methods, such as graphing the inequalities on a coordinate plane.

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

Solving systems of equations has numerous applications in various fields, including physics, engineering, computer science, economics, and biology. It is an essential tool for modeling real-world problems and making predictions.

Q: What are some common mistakes to avoid when solving systems of equations?

Some common mistakes to avoid when solving systems of equations include:

Not checking your work: Failing to check your work can lead to incorrect solutions.
Not using the correct method: Using the wrong method can lead to incorrect solutions.
Not considering all possible solutions: Failing to consider all possible solutions can lead to incorrect conclusions.
Not using technology: Failing to use technology can lead to errors and inconsistencies.

Q: How can I practice solving systems of equations?

There are several ways to practice solving systems of equations, including:

Using online resources: Websites, such as Khan Academy and Mathway, offer interactive lessons and exercises to help you practice solving systems of equations.
Working with a tutor: A tutor can provide one-on-one instruction and help you practice solving systems of equations.
Solving problems from textbooks: Textbooks, such as algebra and geometry textbooks, often include problems that involve solving systems of equations.

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

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

Physics and engineering: Systems of equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
Computer science: Systems of equations are used in computer graphics, game development, and machine learning.
Economics: Systems of equations are used to model economic systems, including supply and demand, inflation, and unemployment.
Biology: Systems of equations are used to model population dynamics, epidemiology, and ecology.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of equations. We have discussed the importance of solving systems of equations, common mistakes to avoid, and real-world applications. We hope this article has been helpful in providing you with a better understanding of solving systems of equations.