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Algebra Worksheets and How To Solve for X

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"Solve for x" is a common phrase in mathematics but where did it come from and what does it mean?

To put it simply, x represents an unknown value in an equation. You need to "solve for x" by finding the constant number that equals x.

The origin of using x to represent an unknown value has been researched by Terry Moore, director of the Radius Foundation in New York, USA.

According to Terry, x was the European translation of an Arabic word for "the unknown thing". This was how Arabic mathematicians represented an unknown value in their equations. For the full explanation see Terry's short and entertaining video.

The medieval Arabs developed algebra from the work of the ancient Babylonians. That means students have been solving for x for the last 25 centuries!

So when a teacher says "solve for x" he or she means:

  1. You will be given an algebraic equation containing one or more constant numbers and a variable number x (by itself or part of a monomial).
    • It's an equation because it has mathematical expressions on both sides of an equals sign.
    • It's algebraic because it uses a letter to represent a variable number.
    • x is a variable number because its numeric value can vary from one equation to another, depending on the constant numbers in the equation.
    • x is part of a monomial if it is multiplied by a constant number, called the coefficient. For example, in the monomial 9x, 9 is the coefficient and x is the variable. Another way to say 9x is "some value equal to nine times another value."
  2. x may appear more than once in the equation, but it is the only variable number.
  3. Your goal is to isolate one and only one x on one side of the equation, and one constant number on the other side.
    • x can end up on the left or right side though we usually arrange it on the left so we can say "x equals ..."
    • The equation is "solved" because you found the numeric value of x by proving x equals a constant number.

Here is an example: 2x = 10

"Solve for x" means to get x by itself on one side of the equation. Using the properties of equality, we can isolate x by dividing both sides of the equation by the coefficient to which x is multiplied. You have to modify both sides of the equation to keep both sides equal.

Let's step through it:

2x = 10
2x / 2 = 10 / 2
1x = 5
x = 5

So x equals 5. Notice how when a number is multiplied by 1, we do not show the 1 any more, because any number multiplied by 1 is equal to itself.

To solve for x when it is divided by a constant, multiply both sides of the equation by the divisor:

x / 3 = 5
x / 3 * 3 = 5 * 3
x = 15

Notice how when two numbers cancel each other out (such as when dividing and multiplying by 3) we just remove those values from the equation.

When x is the divisor, multiply both sides by x:

10 / x = 5
10 / x * x = 5 * x
10 = 5x
10 / 5 = 5x / 5
2 = x

We can complicate the equation by adding a constant number to the variable number. Then you have to subtract the constant from both sides of the equation before solving with division:

2x + 6 = 10
2x + 6 - 6 = 10 - 6
2x = 4
2x / 2 = 4 / 2
1x = 2
x = 2

When a constant value is subtracted, just add it to both sides:

2x - 6 = 10
2x - 6 + 6 = 10 + 6
2x = 16
2x / 2 = 16 / 2
1x = 8
x = 8

Finally, you can complicate the equation even more by having x on both sides of the equation. In that case you have to isolate the variables on one side of the equals sign, then combine them using monomial addition or subtraction. For example:
10x - 30 = 2 - 6x
10x - 30 + 6x = 2 - 6x + 6x
(10 + 6)x - 30 = 2
16x - 30 = 2
16x - 30 + 30 = 2 + 30
16x = 32
16x / 16 = 32 / 16
1x = 2
x = 2

So remember, to solve for x:

  1. Use addition or subtraction to isolate the x monomial.
  2. Use division to isolate x from its coefficient.
  3. Combine and simplify the constant numbers on both sides of the equation.
  4. Congratulations, you just solved for x! The ancient Babylonians would be proud of you.

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Solve for x:
Include multiplication
Example: 2x = 10
Include addition
Example: 10x + 2 = -8x + 2
Include variable on right-hand side
Example: 10x + 2 = -8x + 2

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