Substituting the First Equation into the Second

Oct 13, 2025 by magicalceylontours.com 24 views

Introduction

Solving a system of linear equations is a fundamental concept in mathematics that involves finding the solution to a set of two or more linear equations. In this article, we will focus on solving a system of two linear equations using the substitution method. We will use the given equations to find the solution by substituting the first equation into the second.

Understanding the Problem

We are given two linear equations:

First Equation: ${$ y = -6$}$
Second Equation: ${$ -9x - 12y = 36$}$

Our goal is to find the solution ${$ (x, y)$}$ by substituting the first equation into the second.

The Substitution Method

The substitution method is a technique used to solve a system of linear equations by substituting the expression for one variable from one equation into the other equation. In this case, we will substitute the expression for ${$ y$}$ from the first equation into the second equation.

Substituting the First Equation into the Second

We will substitute ${$ y = -6$}$ into the second equation:

${$ -9x - 12y = 36$}$

Substituting ${$ y = -6$}$ into the equation, we get:

${$ -9x - 12(-6) = 36$}$

Simplifying the Equation

Now, we will simplify the equation by evaluating the expression:

${$ -9x + 72 = 36$}$

Isolating the Variable

Next, we will isolate the variable ${$ x$}$ by subtracting 72 from both sides of the equation:

${$ -9x = 36 - 72$}$

${$ -9x = -36$}$

Solving for x

Now, we will solve for ${$ x$}$ by dividing both sides of the equation by -9:

${$ x = \frac{-36}{-9}$}$

${$ x = 4$}$

Finding the Value of y

Now that we have found the value of ${$ x$}$, we can substitute it into the first equation to find the value of ${$ y$}$:

${$ y = -6$}$

The Solution

Therefore, the solution to the system of linear equations is:

${$ (x, y) = (4, -6)$}$

Conclusion

In this article, we used the substitution method to solve a system of two linear equations. We substituted the expression for one variable from one equation into the other equation and simplified the resulting equation to find the solution. This technique is useful for solving systems of linear equations and is an essential concept in mathematics.

Real-World Applications

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

Physics and Engineering: Solving systems of linear equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Computer Science: Solving systems of linear equations is used in computer graphics, game development, and machine learning.
Economics: Solving systems of linear equations is used to model economic systems, such as supply and demand curves.

Tips and Tricks

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

Use the substitution method: The substitution method is a powerful technique for solving systems of linear equations.
Simplify the equation: Simplifying the equation can make it easier to solve.
Check your work: Always check your work to ensure that the solution is correct.

Common Mistakes

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

Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
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.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. The substitution method is a powerful technique for solving systems of linear equations, and by following the tips and tricks outlined in this article, you can become proficient in solving systems of linear equations.

Introduction

Solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. In this article, we will answer some of the most frequently asked questions about solving systems 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 are solved simultaneously to find the solution.

Q: What are the different methods for solving systems of linear equations?

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

Substitution method: This method involves substituting the expression for one variable from one equation into the other equation.
Elimination method: This method involves eliminating one variable by adding or subtracting the equations.
Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of linear equations by substituting the expression for one variable from one equation into the other equation.

Q: How do I use the substitution method?

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

Solve one equation for one variable: Solve one equation for one variable, such as ${$ y$}$.
Substitute the expression into the other equation: Substitute the expression for the variable into the other equation.
Simplify the equation: Simplify the equation to find the solution.

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of linear equations by eliminating one variable by adding or subtracting the equations.

Q: How do I use the elimination method?

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

Multiply the equations by necessary multiples: Multiply the equations by necessary multiples to make the coefficients of one variable the same.
Add or subtract the equations: Add or subtract the equations to eliminate one variable.
Solve for the other variable: Solve for the other variable to find the solution.

Q: What is the graphical method?

A: The graphical method is a technique used to solve a system of linear equations by graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I use the graphical method?

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

Graph the equations: Graph the equations on a coordinate plane.
Find the point of intersection: Find the point of intersection of the two lines.
Read the solution: Read the solution from the point of intersection.

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

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

Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
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.

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

A: To check your work when solving systems of linear equations, follow these steps:

Plug in the solution: Plug in the solution into both equations to check if it is true.
Check if the solution satisfies both equations: Check if the solution satisfies both equations.
Verify the solution: Verify the solution by checking if it is correct.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the different methods for solving systems of linear equations and avoiding common mistakes, you can become proficient in solving systems of linear equations.