Not expressing one variable in terms of the other

Oct 13, 2025 by windailys.com 56 views

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Introduction

Solving a system of equations by substitution is a powerful technique used in algebra to find the solution to a system of linear equations. This method involves expressing one variable in terms of the other and then substituting it into the other equation to solve for the remaining variable. In this article, we will explore how to solve a system of equations by substitution using a step-by-step approach.

Step 1: Understand the Problem

The problem presents a system of two linear equations with two variables, x and y. The first equation is 6x - 2y = -4, and the second equation is y = -6x + 20. Our goal is to find the values of x and y that satisfy both equations simultaneously.

Step 2: Express One Variable in Terms of the Other

We can start by using the second equation to express y in terms of x. The second equation is y = -6x + 20, which means that y is equal to -6 times x plus 20. We can rewrite this equation as y = -6x + 20.

Step 3: Substitute the Expression into the First Equation

Now that we have expressed y in terms of x, we can substitute this expression into the first equation. The first equation is 6x - 2y = -4. We can replace y with -6x + 20, which gives us 6x - 2(-6x + 20) = -4.

Step 4: Simplify the Equation

To simplify the equation, we can start by distributing the -2 to the terms inside the parentheses. This gives us 6x + 12x - 40 = -4. We can then combine like terms by adding 6x and 12x to get 18x - 40 = -4.

Step 5: Solve for x

Next, we can add 40 to both sides of the equation to get 18x = 36. Then, we can divide both sides by 18 to solve for x. This gives us x = 2.

Step 6: Find the Value of y

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We can use the second equation, y = -6x + 20, and substitute x = 2 to get y = -6(2) + 20. This simplifies to y = -12 + 20, which gives us y = 8.

Step 7: Check the Solution

To check our solution, we can substitute x = 2 and y = 8 into both original equations to make sure they are true. For the first equation, 6x - 2y = -4, we get 6(2) - 2(8) = -4, which simplifies to 12 - 16 = -4. This is true, so our solution is correct.

Conclusion

In this article, we have demonstrated how to solve a system of equations by substitution using a step-by-step approach. We started by expressing one variable in terms of the other, then substituted the expression into the other equation to solve for the remaining variable. We found that the solution to the system is x = 2 and y = 8.

Final Answer

The correct answer is A. (2, 8).

Common Mistakes

When solving a system of equations by substitution, it's easy to make mistakes. Here are some common mistakes to avoid:

Not expressing one variable in terms of the other: Make sure to express one variable in terms of the other before substituting it into the other equation.
Not simplifying the equation: Make sure to simplify the equation after substituting the expression into the other equation.
Not checking the solution: Make sure to check the solution by substituting the values into both original equations to make sure they are true.

Tips and Tricks

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

Use the substitution method when one equation is already solved for one variable: If one equation is already solved for one variable, use the substitution method to solve the system.
Use the elimination method when the coefficients of one variable are the same: If the coefficients of one variable are the same in both equations, use the elimination method to solve the system.
Check the solution by substituting the values into both original equations: Make sure to check the solution by substituting the values into both original equations to make sure they are true.

Real-World Applications

Solving systems of equations by substitution has many real-world applications. Here are a few examples:

Physics and engineering: Solving systems of equations is used to model real-world problems in physics and engineering, such as motion, forces, and energy.
Economics: Solving systems of equations is used to model real-world problems in economics, such as supply and demand, and resource allocation.
Computer science: Solving systems of equations is used to model real-world problems in computer science, such as network flow and optimization problems.

Conclusion

In conclusion, solving a system of equations by substitution is a powerful technique used in algebra to find the solution to a system of linear equations. By expressing one variable in terms of the other and substituting it into the other equation, we can solve for the remaining variable. We have demonstrated how to solve a system of equations by substitution using a step-by-step approach and have provided tips and tricks to help you solve systems of equations by substitution.

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Introduction

Solving a system of equations by substitution is a powerful technique used in algebra to find the solution to a system of linear equations. In our previous article, we demonstrated how to solve a system of equations by substitution using a step-by-step approach. In this article, we will answer some frequently asked questions about solving systems of equations by substitution.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of linear equations by expressing one variable in terms of the other and then substituting it into the other equation to solve for the remaining variable.

Q: When should I use the substitution method?

A: You should use the substitution method when one equation is already solved for one variable, or when the coefficients of one variable are the same in both equations.

Q: How do I express one variable in terms of the other?

A: To express one variable in terms of the other, you can use the second equation to solve for one variable in terms of the other. For example, if the second equation is y = -6x + 20, you can express y in terms of x by rewriting the equation as y = -6x + 20.

Q: What if I have two equations with two variables, but neither equation is solved for one variable?

A: If neither equation is solved for one variable, you can use the elimination method to solve the system. The elimination method involves adding or subtracting the equations to eliminate one variable and solve for the other variable.

Q: How do I check my solution?

A: To check your solution, you can substitute the values into both original equations to make sure they are true. If the values satisfy both equations, then your solution is correct.

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:

Not expressing one variable in terms of the other
Not simplifying the equation after substituting the expression into the other equation
Not checking the solution by substituting the values into both original equations

Q: What are some tips and tricks to help me solve systems of equations by substitution?

A: Some tips and tricks to help you solve systems of equations by substitution include:

Using the substitution method when one equation is already solved for one variable
Using the elimination method when the coefficients of one variable are the same in both equations
Checking the solution by substituting the values into both original equations

Q: How do I apply the substitution method to real-world problems?

A: The substitution method can be applied to real-world problems in physics, engineering, economics, and computer science. For example, in physics, the substitution method can be used to model motion, forces, and energy. In economics, the substitution method can be used to model supply and demand, and resource allocation.

Q: What are some examples of systems of equations that can be solved by substitution?

A: Some examples of systems of equations that can be solved by substitution include:

2x + 3y = 7
y = -2x + 5
4x - 2y = 12
y = 2x - 3

Conclusion

In conclusion, solving a system of equations by substitution is a powerful technique used in algebra to find the solution to a system of linear equations. By expressing one variable in terms of the other and substituting it into the other equation, we can solve for the remaining variable. We have answered some frequently asked questions about solving systems of equations by substitution and provided tips and tricks to help you solve systems of equations by substitution.

Final Answer

The correct answer is A. (2, 8).

Common Mistakes

When solving a system of equations by substitution, it's easy to make mistakes. Here are some common mistakes to avoid:

Not expressing one variable in terms of the other: Make sure to express one variable in terms of the other before substituting it into the other equation.
Not simplifying the equation: Make sure to simplify the equation after substituting the expression into the other equation.
Not checking the solution: Make sure to check the solution by substituting the values into both original equations to make sure they are true.

Tips and Tricks

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

Use the substitution method when one equation is already solved for one variable: If one equation is already solved for one variable, use the substitution method to solve the system.
Use the elimination method when the coefficients of one variable are the same: If the coefficients of one variable are the same in both equations, use the elimination method to solve the system.
Check the solution by substituting the values into both original equations: Make sure to check the solution by substituting the values into both original equations to make sure they are true.

Real-World Applications

Solving systems of equations by substitution has many real-world applications. Here are a few examples:

Physics and engineering: Solving systems of equations is used to model real-world problems in physics and engineering, such as motion, forces, and energy.
Economics: Solving systems of equations is used to model real-world problems in economics, such as supply and demand, and resource allocation.
Computer science: Solving systems of equations is used to model real-world problems in computer science, such as network flow and optimization problems.