Assignment: SubstitutionThe Value Of $x$ In This System Of Equations Is 1.$ \begin{array}{l} 3x + Y = 9 \\ y = -4x + 10 \end{array} $1. Substitute The Value Of $y$ In The First Equation:$ 3x + (-4x + 10) = 9 $2.

Introduction

Systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the concept of substitution, a method used to solve systems of equations. We will use a specific example to illustrate the steps involved in solving a system of equations using substitution.

The Problem

We are given a system of two linear equations:

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

The value of $x$ in this system is given as 1. Our goal is to find the value of $y$.

Step 1: Substitute the Value of $y$ in the First Equation

To solve for $y$, we need to substitute the value of $y$ from the second equation into the first equation. We will replace $y$ with $-4x + 10$ in the first equation.

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

Simplifying the Equation

Now, let's simplify the equation by combining like terms.

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

$$-x + 10 = 9$$

Isolating the Variable

Next, we need to isolate the variable $x$. We can do this by subtracting 10 from both sides of the equation.

$$-x = -1$$

Solving for $x$

Now, we can solve for $x$ by multiplying both sides of the equation by -1.

$$x = 1$$

Discussion

The value of $x$ in the system of equations is indeed 1, as given in the problem. However, we are interested in finding the value of $y$. To do this, we need to substitute the value of $x$ into one of the original equations.

Step 2: Substitute the Value of $x$ into the Second Equation

We will substitute the value of $x$ into the second equation.

$$y = -4(1) + 10$$

Simplifying the Equation

Now, let's simplify the equation by multiplying -4 and 1.

$$y = -4 + 10$$

$$y = 6$$

Conclusion

In this article, we used the method of substitution to solve a system of equations. We started by substituting the value of $y$ from the second equation into the first equation. We then simplified the equation and isolated the variable $x$. Finally, we substituted the value of $x$ into the second equation to find the value of $y$. The value of $y$ is 6.

Why is Substitution Important?

Substitution is an important method for solving systems of equations because it allows us to eliminate one of the variables and solve for the other. This method is particularly useful when one of the equations is linear and the other is quadratic or higher degree.

Real-World Applications

Systems of equations have many real-world applications, including:

• Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
• Economics: Systems of equations are used to model economic systems, including supply and demand curves.
• Computer Science: Systems of equations are used in computer science to solve problems in graph theory and network analysis.

Tips and Tricks

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

• Make sure to substitute the correct value of the variable: It's easy to make mistakes when substituting values, so make sure to double-check your work.
• Simplify the equation carefully: Simplifying the equation can be tricky, so make sure to combine like terms carefully.
• Check your solution: Once you've solved the system of equations, make sure to check your solution by plugging it back into the original equations.

Conclusion

Q: What is substitution in the context of solving systems of equations?

A: Substitution is a method used to solve systems of equations by substituting the value of one variable into the other equation. This allows us to eliminate one of the variables and solve for the other.

Q: When should I use substitution to solve a system of equations?

A: You should use substitution when one of the equations is linear and the other is quadratic or higher degree. Substitution is also useful when one of the variables is already isolated in one of the equations.

Q: How do I know which equation to substitute into?

A: You should substitute the equation that has the variable you want to solve for. For example, if you want to solve for x, you should substitute the equation that has x isolated.

Q: What if I have a system of equations with two variables and two equations, but neither equation has the variable isolated?

A: In this case, you can use the method of elimination to solve the system of equations. This involves adding or subtracting the two equations to eliminate one of the variables.

Q: Can I use substitution to solve a system of equations with more than two variables?

A: Yes, you can use substitution to solve a system of equations with more than two variables. However, this can become complex and may require the use of matrices or other advanced techniques.

Q: What are some common mistakes to avoid when using substitution to solve a system of equations?

A: Some common mistakes to avoid include:

• Not simplifying the equation carefully: Make sure to combine like terms and simplify the equation carefully to avoid errors.
• Not checking the solution: Make sure to check your solution by plugging it back into the original equations to ensure that it is correct.
• Not using the correct value of the variable: Make sure to substitute the correct value of the variable into the other equation.

Q: How can I practice using substitution to solve systems of equations?

A: You can practice using substitution to solve systems of equations by working through examples and exercises. You can also use online resources or math software to help you practice and get feedback on your work.

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

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

• Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
• Economics: Systems of equations are used to model economic systems, including supply and demand curves.
• Computer Science: Systems of equations are used in computer science to solve problems in graph theory and network analysis.

Q: Can I use substitution to solve a system of equations with non-linear equations?

A: Yes, you can use substitution to solve a system of equations with non-linear equations. However, this may require the use of advanced techniques, such as implicit differentiation or numerical methods.

Q: What are some tips for using substitution to solve systems of equations?

A: Some tips for using substitution to solve systems of equations include:

• Make sure to simplify the equation carefully: Simplifying the equation can be tricky, so make sure to combine like terms carefully.
• Check your solution: Once you've solved the system of equations, make sure to check your solution by plugging it back into the original equations.
• Use the correct value of the variable: Make sure to substitute the correct value of the variable into the other equation.

Conclusion

In conclusion, substitution is a powerful method for solving systems of equations. By substituting the value of one variable into the other equation, we can eliminate one of the variables and solve for the other. With practice and patience, you can master the art of substitution and become a proficient problem-solver.