Equations

Axioms of Equality

For all real numbers a, b, and c, the following are some basic rules for using the equal sign.

• Reflexive axiom: a = a.

  
  

  
  

• Symmetric axiom: If a = b, then b = a.

  
  

  
  

• Transitive axiom: If a = b and b = c, then a = c.

  
  

  
  

• Additive axiom: If a = b and c = d, then a + c = b + d.

  
  

  
  

• Multiplicative axiom: If a = b and c = d, then ac = bd.

  
  

  
  

Solving Equations

Remember that an equation is like a balance scale with the equal sign (=) being the fulcrum, or center. Thus, if you do the same thing to both sides of the equal sign (say, add 5 to each side), the equation will still be balanced.

Example 1: Solve for x.

To solve the equation x − 5 = 23, you must get x by itself on one side; therefore, add 5 to both sides.

In the same manner, you may subtract, multiply, or divide both sides of an equation by the same (nonzero) number, and the equation will not change. Sometimes you may have to use more than one step to solve for an unknown.

Example 2: Solve for x.

Subtract 4 from both sides to get the 3 x by itself on one side

Then divide both sides by 3 to get x.

Remember that solving an equation is using opposite operations until the letter is on a side by itself (for addition, subtract; for multiplication, divide, and so forth).

To check, substitute your answer into the original equation.

Example 3: Solve for x.

Add 4 to both sides.

Multiply both sides by 5 to get x.

Example 4: Solve for x.

Add 6 to each side.

Multiply each side by 5/3 (same as dividing by 3/5).

Example 5: Solve for x.

Add −2 x to each side.

Divide both sides by 3.

Example 6: Solve for x.

Add −3 to each side.

Add −4 x to each side.

Divide each side by 2.

Literal Equations

Literal equations have no numbers, only symbols (letters).

Example 7: Solve for Q.

First add X to both sides.

Then divide both sides by P.

Operations opposite to those in the original equation were used to isolate Q. (To remove the − X, a + X was added to both sides of the equation. Because the problem has Q times P, both sides were divided by P.

Example 8: Solve for y.

Multiply both sides by x to get y alone.

Example 9: Solve for x.

To solve this equation quickly, you cross multiply. To cross multiply,

1. Bring the denominators up next to the opposite side numerators and

2. Multiply

   
   

   
   

Then divide both sides by p to get x alone.

Cross multiplying can be used only when the format is two fractions separated by an equal sign.

Be aware that cross multiplying is most effective only when the letter you are solving for is on the bottom (the denominator) of a fraction. If it is on top (the numerator), it is easier simply to clear denominator under the unknown you're solving for.

Example 10: Solve for x.

Multiply both sides by k.

In this problem, there is no need to cross multiply.