Matching Systems with Solutions

Oct 13, 2025 by pynkocean.com 22 views

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore the process of solving systems of linear equations, focusing on the method of substitution and elimination. We will also provide a comprehensive guide to help you match each system on the left with the number of solutions it has on the right.

What are Systems of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. For example:

2x + 3y = 5
x - 2y = -3

Method of Substitution

The method of substitution is a technique used to solve systems of linear equations. This method involves solving one equation for one variable and then substituting that expression into the other equation. For example:

Example 1: Solving a System of Linear Equations using Substitution

Consider the following system of linear equations:

{ \begin{array}{l} x = y - 3 \\ 2x - y = -5 \end{array} \}

To solve this system using substitution, we can start by solving the first equation for x:

x = y - 3

Next, we can substitute this expression for x into the second equation:

2(x) - y = -5

Substituting x = y - 3 into the second equation, we get:

2(y - 3) - y = -5

Expanding and simplifying, we get:

2y - 6 - y = -5

Combine like terms:

y - 6 = -5

Add 6 to both sides:

y = 1

Now that we have found the value of y, we can substitute this value back into one of the original equations to find the value of x. Using the first equation, we get:

x = y - 3 x = 1 - 3 x = -2

Therefore, the solution to this system of linear equations is x = -2 and y = 1.

Example 2: Solving a System of Linear Equations using Substitution

Consider the following system of linear equations:

{ \begin{array}{l} 5x + 2y = -7 \\ 3x - 2y = 2 \end{array} \}

To solve this system using substitution, we can start by solving one of the equations for one variable. Let's solve the first equation for x:

5x + 2y = -7

Subtract 2y from both sides:

5x = -7 - 2y

Divide both sides by 5:

x = (-7 - 2y) / 5

Next, we can substitute this expression for x into the second equation:

3x - 2y = 2

Substituting x = (-7 - 2y) / 5 into the second equation, we get:

3((-7 - 2y) / 5) - 2y = 2

Expanding and simplifying, we get:

(-21 - 6y) / 5 - 2y = 2

Multiply both sides by 5 to eliminate the fraction:

-21 - 6y - 10y = 10

Combine like terms:

-21 - 16y = 10

Add 21 to both sides:

-16y = 31

Divide both sides by -16:

y = -31 / 16

Now that we have found the value of y, we can substitute this value back into one of the original equations to find the value of x. Using the first equation, we get:

5x + 2y = -7

Substituting y = -31 / 16 into the first equation, we get:

5x + 2(-31 / 16) = -7

Expanding and simplifying, we get:

5x - 31 / 8 = -7

Multiply both sides by 8 to eliminate the fraction:

40x - 31 = -56

Add 31 to both sides:

40x = -25

Divide both sides by 40:

x = -25 / 40

x = -5 / 8

Therefore, the solution to this system of linear equations is x = -5 / 8 and y = -31 / 16.

Method of Elimination

The method of elimination is another technique used to solve systems of linear equations. This method involves adding or subtracting the equations to eliminate one of the variables. For example:

Example 3: Solving a System of Linear Equations using Elimination

Consider the following system of linear equations:

{ \begin{array}{l} x + 2y = 3 \\ 2x - 3y = -3 \end{array} \}

To solve this system using elimination, we can start by adding the two equations together:

(x + 2y) + (2x - 3y) = 3 + (-3)

Combine like terms:

3x - y = 0

Next, we can solve this resulting equation for x:

3x = y

Divide both sides by 3:

x = y / 3

Now that we have found the value of x, we can substitute this value back into one of the original equations to find the value of y. Using the first equation, we get:

x + 2y = 3

Substituting x = y / 3 into the first equation, we get:

(y / 3) + 2y = 3

Expanding and simplifying, we get:

y / 3 + 2y = 3

Multiply both sides by 3 to eliminate the fraction:

y + 6y = 9

Combine like terms:

7y = 9

Divide both sides by 7:

y = 9 / 7

Now that we have found the value of y, we can substitute this value back into the equation x = y / 3 to find the value of x:

x = y / 3 x = (9 / 7) / 3 x = 3 / 7

Therefore, the solution to this system of linear equations is x = 3 / 7 and y = 9 / 7.

Conclusion

Solving systems of linear equations is a crucial skill for students and professionals alike. In this article, we have explored the process of solving systems of linear equations using the method of substitution and elimination. We have also provided a comprehensive guide to help you match each system on the left with the number of solutions it has on the right.

Matching Systems with Solutions

Now that we have explored the process of solving systems of linear equations, let's match each system on the left with the number of solutions it has on the right.

System 1:

{ \begin{array}{l} x = y - 3 \\ 2x - y = -5 \end{array} \}

This system has 1 solution.

System 2:

{ \begin{array}{l} 5x + 2y = -7 \\ 3x - 2y = 2 \end{array} \}

This system has 1 solution.

System 3:

{ \begin{array}{l} x + 2y = 3 \\ 2x - 3y = -3 \end{array} \}

This system has 1 solution.

Final Thoughts

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.

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

A: The two main methods for solving systems of linear equations are the method of substitution and the method of elimination.

Q: What is the method of substitution?

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I know which method to use?

A: You can use the following steps to determine which method to use:

If one of the equations is already solved for one variable, use the method of substitution.
If the coefficients of one variable are the same in both equations, but with opposite signs, use the method of elimination.
If the coefficients of one variable are not the same in both equations, but you can multiply one or both equations by a constant to make the coefficients the same, use the method of elimination.

Q: What is the difference between a dependent and an independent system?

A: A dependent system is a system of linear equations that has an infinite number of solutions. An independent system is a system of linear equations that has a unique solution.

Q: How do I determine if a system is dependent or independent?

A: You can use the following steps to determine if a system is dependent or independent:

If the two equations are identical, the system is dependent.
If the two equations are not identical, but one equation is a multiple of the other, the system is dependent.
If the two equations are not identical and not multiples of each other, the system is independent.

Q: What is the difference between a consistent and an inconsistent system?

A: A consistent system is a system of linear equations that has at least one solution. An inconsistent system is a system of linear equations that has no solution.

Q: How do I determine if a system is consistent or inconsistent?

A: You can use the following steps to determine if a system is consistent or inconsistent:

If the system has at least one solution, the system is consistent.
If the system has no solution, the system is inconsistent.

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 following the order of operations when simplifying expressions.
Not checking for extraneous solutions.
Not using the correct method for solving the system.
Not checking the final answer for consistency.

Q: How can I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by:

Working through examples and exercises in a textbook or online resource.
Using online tools or software to generate random systems of linear equations.
Creating your own systems of linear equations and solving them.
Joining a study group or working with a tutor to practice solving systems of linear equations.

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

A: Some real-world applications of solving systems of linear equations include:

Finding the intersection of two lines or curves.
Determining the cost of producing a product based on the number of units produced and the cost per unit.
Calculating the amount of money in a bank account based on the interest rate and the number of years the money has been invested.
Determining the amount of time it will take to complete a project based on the number of tasks and the time required to complete each task.

Conclusion

Solving systems of linear equations is a crucial skill for students and professionals alike. By understanding the process of solving systems of linear equations using the method of substitution and elimination, you can tackle even the most complex systems with confidence. Remember to always follow the order of operations, check for extraneous solutions, and use the correct method for solving the system. With practice and patience, you can become proficient in solving systems of linear equations and apply this skill to real-world problems.