Writing an Equivalent System of Equations

Oct 13, 2025 by miodeya.com 61 views

Equivalent systems of equations are sets of linear equations that have the same solution set. In other words, if two systems of equations have the same set of solutions, then they are equivalent. In this article, we will explore how to write an equivalent system of equations and understand the concept of equivalent systems.

What are Equivalent Systems of Equations?

Equivalent systems of equations are sets of linear equations that have the same solution set. This means that if we have two systems of equations, and they have the same set of solutions, then they are equivalent. For example, consider the following two systems of equations:

a.$${ \begin{array}{l} x + 2y = 4 \ 2x - 5y = -1 \end{array} }$$

b.$${ \begin{array}{l} x + 2y = 4 \ -10x + 25y = 10 \end{array} }$$

These two systems of equations are equivalent because they have the same set of solutions. To see this, we can solve one of the systems and then substitute the solution into the other system. Let's solve system (a) and then substitute the solution into system (b).

Solving System (a)

To solve system (a), we can use the method of substitution or elimination. Let's use the elimination method. We can multiply the first equation by 2 and the second equation by 1, and then add the two equations to eliminate the variable x.

Multiplying the first equation by 2, we get:

2(x + 2y) = 2(4)

Expanding and simplifying, we get:

2x + 4y = 8

Multiplying the second equation by 1, we get:

2x - 5y = -1

Adding the two equations, we get:

(2x + 4y) + (2x - 5y) = 8 + (-1)

Simplifying, we get:

4x - y = 7

Now we have a new equation with one variable. We can solve for x by adding y to both sides of the equation:

4x = 7 + y

Dividing both sides by 4, we get:

x = (7 + y)/4

Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's substitute it into the first equation:

x + 2y = 4

Substituting x = (7 + y)/4, we get:

(7 + y)/4 + 2y = 4

Multiplying both sides by 4, we get:

7 + y + 8y = 16

Combine like terms:

9y = 9

Dividing both sides by 9, we get:

y = 1

Now that we have the value of y, we can substitute it into the equation x = (7 + y)/4 to solve for x:

x = (7 + 1)/4

x = 8/4

x = 2

So, the solution to system (a) is x = 2 and y = 1.

Substituting the Solution into System (b)

Now that we have the solution to system (a), we can substitute it into system (b) to see if it is equivalent. Let's substitute x = 2 and y = 1 into the second equation of system (b):

-10x + 25y = 10

Substituting x = 2 and y = 1, we get:

-10(2) + 25(1) = 10

Expanding and simplifying, we get:

-20 + 25 = 10

Simplifying further, we get:

5 = 10

This is a contradiction, which means that system (b) is not equivalent to system (a).

Conclusion

In this article, we explored the concept of equivalent systems of equations. We saw that two systems of equations are equivalent if they have the same set of solutions. We also saw how to write an equivalent system of equations by solving one of the systems and then substituting the solution into the other system. Finally, we saw that not all systems of equations are equivalent, and that we can use the method of substitution or elimination to determine if two systems are equivalent.

Writing an Equivalent System of Equations

To write an equivalent system of equations, we can follow these steps:

Solve one of the systems of equations using the method of substitution or elimination.
Substitute the solution into the other system of equations.
Simplify the resulting equations to get the equivalent system.

For example, consider the following system of equations:

{ \begin{array}{l} x + 2y = 4 \\ 2x - 5y = -1 \end{array} \}

To write an equivalent system of equations, we can solve this system using the method of elimination. We can multiply the first equation by 2 and the second equation by 1, and then add the two equations to eliminate the variable x.

Multiplying the first equation by 2, we get:

2(x + 2y) = 2(4)

Expanding and simplifying, we get:

2x + 4y = 8

Multiplying the second equation by 1, we get:

2x - 5y = -1

Adding the two equations, we get:

(2x + 4y) + (2x - 5y) = 8 + (-1)

Simplifying, we get:

4x - y = 7

Now we have a new equation with one variable. We can solve for x by adding y to both sides of the equation:

4x = 7 + y

Dividing both sides by 4, we get:

x = (7 + y)/4

Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's substitute it into the first equation:

x + 2y = 4

Substituting x = (7 + y)/4, we get:

(7 + y)/4 + 2y = 4

Multiplying both sides by 4, we get:

7 + y + 8y = 16

Combine like terms:

9y = 9

Dividing both sides by 9, we get:

y = 1

Now that we have the value of y, we can substitute it into the equation x = (7 + y)/4 to solve for x:

x = (7 + 1)/4

x = 8/4

x = 2

So, the solution to the system is x = 2 and y = 1.

Substituting the Solution into the Other System

Now that we have the solution to the system, we can substitute it into the other system of equations to get the equivalent system. Let's substitute x = 2 and y = 1 into the other system of equations:

{ \begin{array}{l} x + 2y = 4 \\ 2x - 5y = -1 \end{array} \}

Substituting x = 2 and y = 1, we get:

2 + 2(1) = 4

Expanding and simplifying, we get:

2 + 2 = 4

Simplifying further, we get:

4 = 4

This is a true statement, which means that the two systems of equations are equivalent.

Conclusion

Writing an Equivalent System of Equations Using the Method of Substitution

To write an equivalent system of equations using the method of substitution, we can follow these steps:

Solve one of the systems of equations using the method of substitution.
Substitute the solution into the other system of equations.
Simplify the resulting equations to get the equivalent system.

For example, consider the following system of equations:

{ \begin{array}{l} x + 2y = 4 \\ 2x - 5y = -1 \end{array} \}

To write an equivalent system of equations using the method of substitution, we can solve this system using the method of substitution. We can solve for x in terms of y by subtracting 2y from both sides of the first equation:

x = 4 - 2y

Now that we have the value of x, we can substitute it into the second equation to solve for y:

2x - 5y = -1

Substituting x = 4 - 2y, we get:

2(4 - 2y) - 5y = -1

Expanding and simplifying, we get:

8 - 4y - 5y = -1

Combine like terms:

-9y = -9

Dividing both sides by -9, we get:

y = 1

Now that we have the value of y, we can substitute it into the equation x = 4 - 2y to solve for x:

x = 4 - 2(1)

x = 4 - 2

x = 2

So, the solution to the system is x = 2