Substituting the Value of y in the First Equation

Oct 13, 2025 by miodeya.com 57 views

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it requires a deep understanding of algebraic manipulations. In this article, we will explore a system of two linear equations and find the value of x that satisfies both equations. We will use a step-by-step approach to solve the system, and we will provide a detailed explanation of each step.

The System of Equations

The system of equations is given by:

$\begin{array}{l} 3x + y = 9 \\ y = -4x + 10 \end{array}$

We are asked to find the value of x that satisfies both equations.

Substituting the Value of y in the First Equation

To solve the system, we can substitute the value of y from the second equation into the first equation. This will give us an equation with only one variable, x.

We will substitute y = -4x + 10 into the first equation:

$3x + (-4x + 10) = 9$

Combining Like Terms

Now, we will combine like terms in the equation:

$3x - 4x + 10 = 9$

Simplifying the Equation

Next, we will simplify the equation by combining the like terms:

$-x + 10 = 9$

Isolating x

Now, we will isolate x by subtracting 10 from both sides of the equation:

$-x = -1$

Solving for x

Finally, we will solve for x by multiplying both sides of the equation by -1:

$x = 1$

Conclusion

In this article, we have solved a system of two linear equations and found the value of x that satisfies both equations. We have used a step-by-step approach to solve the system, and we have provided a detailed explanation of each step. The value of x in this system of equations is 1.

Why is Solving Systems of Equations Important?

Solving systems of equations is an important concept in mathematics because it has many real-world applications. For example, in physics, systems of equations are used to model the motion of objects. In economics, systems of equations are used to model the behavior of markets. In engineering, systems of equations are used to design and optimize systems.

How to Solve Systems of Equations

There are several methods to solve systems of equations, including:

Substitution Method: This method involves substituting the value of one variable into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Tips and Tricks

Here are some tips and tricks to help you solve systems of equations:

Read the problem carefully: Make sure you understand what the problem is asking for.
Use a step-by-step approach: Break down the problem into smaller steps and solve each step separately.
Check your work: Make sure you have solved the problem correctly by checking your work.

Real-World Applications

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

Physics: Systems of equations are used to model the motion of objects.
Economics: Systems of equations are used to model the behavior of markets.
Engineering: Systems of equations are used to design and optimize systems.

Conclusion

In conclusion, solving systems of equations is an important concept in mathematics that has many real-world applications. We have used a step-by-step approach to solve a system of two linear equations and found the value of x that satisfies both equations. We have also provided a detailed explanation of each step and highlighted the importance of solving systems of equations.

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it can be a challenging topic for many students. In this article, we will answer some of the most frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other. Each equation in the system is a statement that two or more variables are equal to a certain value.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including the substitution method, elimination method, and graphical method. The substitution method involves substituting the value of one variable into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one variable.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of equations by substituting the value of one variable into the other equation. This method is useful when one of the equations is already solved for one of the variables.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of equations by adding or subtracting the equations to eliminate one variable. This method is useful when the coefficients of the variables in the two equations are the same.

Q: What is the graphical method?

A: The graphical method is a method of solving a system of equations by graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I know which method to use?

A: The choice of method depends on the type of system of equations and the variables involved. If the system is a linear system with two variables, the substitution or elimination method may be the best choice. If the system is a nonlinear system, the graphical method may be the best choice.

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

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

Not checking the work: Make sure to check the work to ensure that the solution is correct.
Not using the correct method: Choose the correct method for the type of system of equations.
Not simplifying the equations: Simplify the equations to make it easier to solve the system.

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

A: To check your work, substitute the solution back into the original equations to ensure that it satisfies both equations.

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

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

Physics: Systems of equations are used to model the motion of objects.
Economics: Systems of equations are used to model the behavior of markets.
Engineering: Systems of equations are used to design and optimize systems.

Q: Can I use a calculator to solve a system of equations?

A: Yes, you can use a calculator to solve a system of equations. However, it is always a good idea to check the work to ensure that the solution is correct.

Conclusion

In conclusion, solving systems of equations is a fundamental concept in mathematics that has many real-world applications. We have answered some of the most frequently asked questions about solving systems of equations and provided some tips and tricks to help you solve systems of equations.