How to Solve Rational Equations
If necessary, rearrange your equation to get one fraction on each side of the equals sign., Cross-multiply., Set the two products as equal to each other., Solve for your variable.
Step-by-Step Guide
-
Step 1: If necessary
Cross-multiplication is a quick, easy way of solving rational equations.
Unfortunately, this method only works for rational equations that contain exactly one rational expression or fraction on each side of the equals sign.
If your equation isn't in proper cross-multiplication form, you may need to use algebraic operations to move its terms into their proper places.
For instance, the equation (x + 3)/4
- x/(-2) = 0 can easily be rearranged into cross-multiplication form by adding x/(-2) to both sides of the equation, leaving you with (x + 3)/4 = x/(-2).
Keep in mind that decimals and whole numbers can be made into fractions by giving them a denominator of
1. (x + 3)/4
-
2.5 = 5, for instance, can be rewritten as (x + 3)/4 =
7.5/1, making it a valid candidate for cross-multiplication.
Some rational equations can't easily be reduced into a form with one fraction or rational equation on each side of the equals sign.
In such cases, use a lowest common denominator approach. -
Step 2: rearrange your equation to get one fraction on each side of the equals sign.
Cross-multiplication simply means multiplying one fraction's numerator by the other's denominator and vice versa.
Multiply the numerator of the fraction on the left of the equal sign by the denominator of the fraction on the right.
Repeat with the numerator of the right-hand fraction and the denominator of the fraction on the left.
Cross-multiplication works according to basic algebraic principals.
Rational expressions and other fractions can be made into non-fractions by multiplying them by their denominators.
Cross-multiplication is basically a handy shortcut for multiplying both sides of the equation by both fraction's denominators.
Don't believe it? Try it
- you'll get the same results after simplifying. , After cross-multiplying, you'll have two products.
Set these two terms equal to each other and simplify to get each side of the equation in its simplest terms.
For example, if your original rational expression was (x+3)/4 = x/(-2), after cross multiplying, your new equation is
-2(x+3) = 4x.
If we wish, this can also be written as
-2x
- 6 = 4x. , Use algebraic operations to solve for the variable in your equation.
Remember that, if x appears on both sides of the equals sign, you'll need to add or subtract x terms to both sides to get x terms on only one side of the equals sign In our example, we can divide both sides of the equation by
-2, giving us x+3 =
-2x.
Subtracting x from both sides gives us 3 =
-3x.
Finally, dividing both sides by
-3 gives us
-1 = x, which we can re-write as x =
-1.
We have found x, solving our rational equation. -
Step 3: Cross-multiply.
-
Step 4: Set the two products as equal to each other.
-
Step 5: Solve for your variable.
Detailed Guide
Cross-multiplication is a quick, easy way of solving rational equations.
Unfortunately, this method only works for rational equations that contain exactly one rational expression or fraction on each side of the equals sign.
If your equation isn't in proper cross-multiplication form, you may need to use algebraic operations to move its terms into their proper places.
For instance, the equation (x + 3)/4
- x/(-2) = 0 can easily be rearranged into cross-multiplication form by adding x/(-2) to both sides of the equation, leaving you with (x + 3)/4 = x/(-2).
Keep in mind that decimals and whole numbers can be made into fractions by giving them a denominator of
1. (x + 3)/4
-
2.5 = 5, for instance, can be rewritten as (x + 3)/4 =
7.5/1, making it a valid candidate for cross-multiplication.
Some rational equations can't easily be reduced into a form with one fraction or rational equation on each side of the equals sign.
In such cases, use a lowest common denominator approach.
Cross-multiplication simply means multiplying one fraction's numerator by the other's denominator and vice versa.
Multiply the numerator of the fraction on the left of the equal sign by the denominator of the fraction on the right.
Repeat with the numerator of the right-hand fraction and the denominator of the fraction on the left.
Cross-multiplication works according to basic algebraic principals.
Rational expressions and other fractions can be made into non-fractions by multiplying them by their denominators.
Cross-multiplication is basically a handy shortcut for multiplying both sides of the equation by both fraction's denominators.
Don't believe it? Try it
- you'll get the same results after simplifying. , After cross-multiplying, you'll have two products.
Set these two terms equal to each other and simplify to get each side of the equation in its simplest terms.
For example, if your original rational expression was (x+3)/4 = x/(-2), after cross multiplying, your new equation is
-2(x+3) = 4x.
If we wish, this can also be written as
-2x
- 6 = 4x. , Use algebraic operations to solve for the variable in your equation.
Remember that, if x appears on both sides of the equals sign, you'll need to add or subtract x terms to both sides to get x terms on only one side of the equals sign In our example, we can divide both sides of the equation by
-2, giving us x+3 =
-2x.
Subtracting x from both sides gives us 3 =
-3x.
Finally, dividing both sides by
-3 gives us
-1 = x, which we can re-write as x =
-1.
We have found x, solving our rational equation.
About the Author
Sara Foster
Committed to making crafts accessible and understandable for everyone.
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