Equivalent equations are algebraic equations having identical solutions or roots. Identifying, solving, and forming equivalent equations is a valuable algebra skill in both the classroom and everyday life. Here are examples of equivalent equations, the rules they follow, how to solve them, and practical applications.

- Equivalent equations have identical solutions.
- Equations with no roots are equivalent.
- Adding or subtracting the same number or expression to both sides of an equation results in equivalent equation.
- Multiplying or dividing both sides of an equation by the same non-zero number forms an equivalent equation.

### Rules for Equivalent Equations

There are several ways to make equivalent equations:

- Adding or subtracting the same number or expression to both sides of an equation forms an equivalent equation.
- Multiplying or dividing both sides of an equation by the same non-zero number forms an equivalent equation.
- Raising both sides of an equation by the same odd power or root produces an equivalent equation. This is because multiplying by an odd number keeps the “sign” the same on both sides of the equation.
- Raising both sides of a non-negative equation to the same even power or root forms an equivalent equation. This does not work with negative equations because it changes the sign.
- Equations are equivalent only if they have exactly the same roots. If one equation has a root another doesn’t have, the equations are not equivalent.

You use these rules simplifying and solving equations. For example, solving x + 1 = 0, you isolate the variable to get the solution. In this case, you subtract “1” from both sides of the equation:

- x + 1 = 0
- x + 1 – 1 = 0 – 1
- x = -1

All of the equations are equivalent.

In solving 2x + 4 = 6x + 12:

- 2x + 4 = 6x + 12
- 2x – 6x + 4 – 4 = 6x – 6x + 12 – 4
- -4x = 8
- -4x/(-4) = 8/(-4)
- x = -2

### Examples of Equivalent Equations

#### Equations With No Variables

Here are examples of equivalent equations without variables:

- 3 + 2 = 5
- 4 + 1 = 5
- 5 + 0 = 5
- -3 + 8 = 10 – 5

These equations are *not* equivalent:

- 3 + 2 = 5
- 4 + 3 = 7

#### Equations With One Variable

These equations are examples of equivalent linear equations with one variable:

- x = 5
- -2x = 10

In both equations, x = 5.

These equations are also equivalent:

- x
^{2}+ 1 = 0 - 2x
^{2}+ 1 = 3

In both cases, x is the square root of -1 or *i*.

These equations are *not* equivalent, because the first equation has two roots (6, -6) and the second equation has one root (6):

- x
^{2}= 36 - x – 6 = 0

#### Equations With Two Variables

Here are two equations with two unknowns (x and y):

- 3x + 12y = 15
- 7x – 10y = -2

These equations are equivalent to this set of equations:

- x + 4y = 5
- 7x – 10y = -2

To verify this, solve for “x” and “y”. If the values are the same for both sets of equations, then they are equivalent.

First, isolate one variable (it does not matter which one) and plug its solution in to the other equation.

- 3x + 12y = 15
- 3x = 15 – 12y
- x = (15 – 12y)/3 = 5 – 4y

Use this value for “x” in the second equation:

- 7x – 10y = -2
- 7(5 – 4 y) – 10y = -2
- 7y – 10y = -2
- -3y = -2
- y = 2/3

Now, use this solution for “y” in the other equation and solve for “x”:

- x + 4y = 5
- x + (4)(2/3) = 5
- x = 5 – (8/3)
- x = (5*3)/3 – 8/3
- x = 15/3 – 8/3
- x = 7/3

Of course, it’s easier if you just recognize that the first equation in the first set is three times the first equation in the second set!

### A Practical Use of Equivalent Equations

You use equivalent equations in daily life. For example, you use them when comparing prices while shopping.

If one company has a shirt for $6 with $12 shipping and another company has the same shirt for $7.50 with $9 shipping, which company offers the better deal? How many shirts do you need to buy for the prices to be the same at both companies?

First, find how much one shirt costs for each company:

- Price #1 = 6x + 12 = (6)(1) + 12 = 6 + 12 = $18
- Price #2 = 7.5x + 9 = (1)(7.5) + 9 = 7.5 + 9 = $16.50

The second company offers the better deal if you’re only getting one shirt. But, use equivalent equations and find how many shirts you need to buy for the other company to be the same price. Set the equations equal to one another and solve for x:

- 6x + 12 = 7.5x + 9
- 6x – 7.5x = 9 – 12 (subtracting the same numbers or expressions from each side)
- -1.5x = -3
- 1.5x = 3 (dividing both sides by the same number, -1)
- x = 3/1.5 (dividing both sides by 1.5)
- x = 2

So, if you buy two shirts, the price plus shipping is the same, no matter which company you choose. Also, if you buy more than two shirts, the first company has the better deal!

### References

- Barnett, R.A.; Ziegler, M.R.; Byleen, K.E. (2008).
*College Mathematics for Business, Economics, Life Sciences and the Social Sciences*(11th ed.). Upper Saddle River, N.J.: Pearson. ISBN 978-0-13-157225-6. - Hosch, William L. (ed.) (2010).
*The Britannica Guide to Algebra and Trigonometry*. Britannica Educational Publishing. The Rosen Publishing Group. ISBN 978161530219. - Kaufmann, Jerome E.; Schwitters, Karen L. (2010).
*Algebra for College Students*. Cengage Learning. ISBN 9780538733540. - Larson, Ron; Hostetler, Robert (2007).
*Precalculus: A Concise Course*. Houghton Mifflin. ISBN 978-0-618-62719-6.