Solving Systems of Radical Equations

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will provide instruction on how to solve systems of radical equations. The definition of a radical equation and systems of radical equations will be provided and examples of how to solve them will be given.

Radical Equations

Did you know that the square root of a number can be called a radical? When we talk about a radical equation we are not using eighties slang, as you might think. Instead, a radical equation is an equation where the variable is under the square root symbol, also called the radical symbol.


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If we have more than one radical equation, with more than one variable, we have a system of radical equations. Let's look at what a system of radical equations looks like.

Solving Systems of Radical Equations


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In order to solve a system of radical equations, we can use the substitution method. To solve using the substitution method, we need to have at least one equation solved for one variable. Since both equations in this example start with y =, they are both solved for y. Basically, this means that y is alone on one side of the equation.

The second step is to replace the variable y with what it is equal to. This replacement of the y variable is the substitution.


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Since the second equation is equal to y, we will replace the y in the first equation with the square root of two times x minus 5.


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Now we have one equation, with one variable. We are now able to solve this equation since there is only one variable.

First, we must get rid of the radicals. If we have multiplication in the equation , to undo that operation, we need to divide. If we have addition, we need to subtract to solve. This is known as an inverse operation, or an operation that undoes another operation. So, how do we undo a radical?

To undo a radical, we will square both sides. The inverse operation of a radical is to square it.


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When we square both sides, all it does is undo the radical, so the radicals go away. The terms underneath the radical do not change.


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Now we can solve this equation, by moving all the variables to one side of the equation and moving the numbers to the other side. We subtract x from both sides. The x on the right will cancel and on the left, 2x minus x is equal to x. Then we need to move the numbers to the right side. We must add five to both sides. The fives on the left side will cancel each other out, and then we add two and five and get x equals seven.


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You might think we are finished because we know what x is equal to, but we started with two equations and two variables. We need to find out what y is equal to.

In order to find y, we can use either equation that we started with. We will simply replace x with seven and solve for y. To show you that it doesn't matter which equation you choose, let's plug seven in for x in both equations.


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Notice, if we choose the first equation, we get y = 3. If we choose the second equation, we also get y = 3.'

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