Solve Each System By Substitution.1) $ \begin{array}{l} y = 1 \ 4x + 4y = 20 \end{array} }$Substitute { Y = 1 $}$ Into The Second Equation ${ 4x + 4(1) = 20 }$Now Solve For { X $ $.

Introduction

Solving systems of equations is a fundamental concept in mathematics, and there are several methods to approach this problem. In this article, we will focus on solving systems of equations by substitution. This method involves substituting one equation into the other to solve for the variables. We will use a step-by-step approach to solve each system of equations.

System 1: Solving by Substitution

Problem

{ \begin{array}{l} y = 1 \\ 4x + 4y = 20 \end{array} \}

Solution

To solve this system by substitution, we will substitute the value of $y$ into the second equation.

Step 1: Substitute $y = 1$ into the second equation

{ 4x + 4(1) = 20 \}

This simplifies to:

{ 4x + 4 = 20 \}

Step 2: Solve for $x$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can do this by subtracting 4 from both sides of the equation.

{ 4x = 20 - 4 \}

This simplifies to:

{ 4x = 16 \}

Step 3: Divide both sides by 4

To solve for $x$, we need to divide both sides of the equation by 4.

{ x = \frac{16}{4} \}

This simplifies to:

{ x = 4 \}

Therefore, the solution to this system of equations is $x = 4$ and $y = 1$.

Discussion

Solving systems of equations by substitution is a powerful method that can be used to solve a wide range of problems. This method involves substituting one equation into the other to solve for the variables. In this example, we used the substitution method to solve a system of two equations with two variables. We substituted the value of $y$ into the second equation and then solved for $x$. This method can be used to solve systems of equations with any number of variables.

System 2: Solving by Substitution

Problem

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

Solution

To solve this system by substitution, we will substitute the value of $x$ into the second equation.

Step 1: Solve the first equation for $x$

{ x + y = 5 \}

This simplifies to:

{ x = 5 - y \}

Step 2: Substitute $x = 5 - y$ into the second equation

{ 2(5 - y) - 2y = -2 \}

This simplifies to:

{ 10 - 2y - 2y = -2 \}

Step 3: Combine like terms

{ 10 - 4y = -2 \}

Step 4: Subtract 10 from both sides

{ -4y = -12 \}

Step 5: Divide both sides by -4

{ y = \frac{-12}{-4} \}

This simplifies to:

{ y = 3 \}

Step 6: Substitute $y = 3$ into the first equation

{ x + 3 = 5 \}

This simplifies to:

{ x = 2 \}

Therefore, the solution to this system of equations is $x = 2$ and $y = 3$.

Discussion

Solving systems of equations by substitution is a powerful method that can be used to solve a wide range of problems. This method involves substituting one equation into the other to solve for the variables. In this example, we used the substitution method to solve a system of two equations with two variables. We substituted the value of $x$ into the second equation and then solved for $y$. This method can be used to solve systems of equations with any number of variables.

Conclusion

Solving systems of equations by substitution is a powerful method that can be used to solve a wide range of problems. This method involves substituting one equation into the other to solve for the variables. We used the substitution method to solve two systems of equations with two variables. We substituted the value of $y$ into the second equation and then solved for $x$ in the first system, and we substituted the value of $x$ into the second equation and then solved for $y$ in the second system. This method can be used to solve systems of equations with any number of variables.

Tips and Tricks

• Make sure to substitute the correct value into the equation.
• Make sure to solve for the correct variable.
• Make sure to check your work by plugging the solution back into the original equations.

Real-World Applications

Solving systems of equations by substitution has many real-world applications. For example, it can be used to solve problems in physics, engineering, and economics. It can also be used to solve problems in computer science, such as solving systems of linear equations.

Conclusion

$$$$$$$$

Introduction

Solving systems of equations by substitution is a powerful method that can be used to solve a wide range of problems. In this article, we will answer some of the most frequently asked questions about solving systems of equations by substitution.

Q: What is solving systems of equations by substitution?

A: Solving systems of equations by substitution is a method of solving systems of equations where one equation is substituted into the other to solve for the variables.

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

A: You can choose either equation to substitute into the other. The key is to make sure that you are substituting the correct value into the equation.

Q: What if I get stuck while solving the system?

A: If you get stuck while solving the system, try going back to the original equations and re-evaluating your work. Make sure that you are following the correct steps and that you are not making any mistakes.

Q: Can I use the substitution method to solve systems of equations with more than two variables?

A: Yes, you can use the substitution method to solve systems of equations with more than two variables. However, it may be more difficult to solve the system and you may need to use additional techniques such as elimination or graphing.

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

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

• Substituting the wrong value into the equation
• Not following the correct steps
• Not checking your work
• Not using the correct method for the type of system you are solving

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

A: To check your work when solving systems of equations by substitution, plug the solution back into the original equations and make sure that it satisfies both equations.

Q: Can I use a calculator to solve systems of equations by substitution?

A: Yes, you can use a calculator to solve systems of equations by substitution. However, it is still important to understand the steps involved in solving the system and to be able to check your work.

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

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

• Physics: Solving systems of equations by substitution can be used to solve problems in physics, such as finding the position and velocity of an object.
• Engineering: Solving systems of equations by substitution can be used to solve problems in engineering, such as designing a bridge or a building.
• Economics: Solving systems of equations by substitution can be used to solve problems in economics, such as finding the equilibrium price and quantity of a good.

Conclusion

Solving systems of equations by substitution is a powerful method that can be used to solve a wide range of problems. By understanding the steps involved in solving the system and by being able to check your work, you can use this method to solve systems of equations with any number of variables.

Tips and Tricks

• Make sure to substitute the correct value into the equation.
• Make sure to solve for the correct variable.
• Make sure to check your work by plugging the solution back into the original equations.
• Use a calculator to check your work and to make sure that you are getting the correct solution.

Real-World Applications

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

• Physics: Solving systems of equations by substitution can be used to solve problems in physics, such as finding the position and velocity of an object.
• Engineering: Solving systems of equations by substitution can be used to solve problems in engineering, such as designing a bridge or a building.
• Economics: Solving systems of equations by substitution can be used to solve problems in economics, such as finding the equilibrium price and quantity of a good.

Conclusion

Solving systems of equations by substitution is a powerful method that can be used to solve a wide range of problems. By understanding the steps involved in solving the system and by being able to check your work, you can use this method to solve systems of equations with any number of variables.