This topic illustrates how the Edit > ObjectEE7FDI and Simplify commandsSimplify_commands can be used to solve equations and relations manually by issuing a sequence of commands.  This procedure is useful for learning how to solve such problems.  Use the Solve > Expression command1PET_XA to automate the process of solving equations and relations.


If an equation is added to an equation, the result is an equation whose left side is the sum of the left sides of the original equations and whose right side is the sum of the rights sides of the original equations.  Similar remarks apply to subtracting, multiplying, and dividing an equation by an equation.


If an expression, other than an equation, is added to an equation, the expression is added to both sides of the equation.  Similar remarks apply to subtracting, multiplying, and dividing an equation by an expression.


The ability to combine equations with expressions makes it possible to solve equations in a step-by-step manner.  For example, to manually solve the linear equation

2·x - 3 = 5

for x, first enter the equation.  Then press the F4 key to copy the equation to the expression entry line enclosed in parentheses and add 3 to both sides of the equation by entering the expression

(2·x - 3 = 5) + 3

which simplifies to

2·x = 8


Then press the F4 key again and divide both sides of the equation by 2 by entering the expression

(2·x = 8)/2

which simplifies to the solution of the original equation

x = 4


Quadratic equations can also be solved manually by completing the square.  For example, to solve the quadratic equation x²+5·x+6=0 for x, start by subtracting 6 from both sides of the equation by entering the expression

(x^2 + 5·x + 6 = 0) - 6

which simplifies to

 2            
x  + 5·x = -6 


Then complete the square by adding 5/2 squared to both sides of the equation by pressing the F4 key and entering the expression

(x^2 + 5·x = -6) + (5/2)^2

which factors to

          2        
 (2·x + 5)      1  
———————————— = ——— 
      4         4  


Then press the F4 key and multiply both sides of the equation by 4 giving

         2     
(2·x + 5)  = 1 


Taking the square root of both sides of this equation giving the two linear equations

2·x + 5 = 1  

2·x + 5 = -1 

that can be solved as described previously.


Quadratic and higher order polynomial equations can be solved by collecting all the terms on one side of the equation and factoring.  For example, rationally factoring the equation

 2               
x  - 5·x + 6 = 0 

yields the equation

(x - 2)·(x - 3) = 0


Then set the factors equal to 0 by highlighting each factor and using the Edit > Object commandEE7FDI.  This gives the two linear equations

x - 2 = 0

x - 3 = 0

that can be solved as described previously.


The result of applying a function to an equation is an equation whose left and right sides are the result of independently applying the function to the left and right sides of the original equation.  For example, to solve the equation

LN(x) = 5

where LN is the natural logarithm function (see Built-in Functions and ConstantsBuilt_in_Functions_and_Constants), use the Edit > Object commandEE7FDI to apply the exponential or inverse logarithm function EXP to the equation.  This gives

EXP(LN(x) = 5)

which simplifies to the solution of the original equation

     5 
x = ê  


Be aware that applying one of the Inverse Trigonometric FunctionsInverse_Trigonometric_Functions to an equation discards all solutions not on the principal branch.  Operations such as clearing the denominator or squaring both sides of an equation can introduce spurious solutions.  For example, squaring both sides of the equation

SQRT(x) = -1

yields x=1, which is not a solution.  Solutions should always be verified by substituting them into the original relation.

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