Solving Equations Graphically, Solutions to Linear Equations

Intro to Linear Equations Linear Equation Solver - The Page Does The Work Solving Equation on Graphing Calculators

Intro

      Three kinds of solutions are possible when one solves an equation:

• ONE OR MORE SOLUTIONS
  - the statement is sometimes true,
  for example, x = 5 or 2 = y, or
  3 + 2x = 1, or 3 + 2x = -1, or 2x + 3 = 4x - 1, or x + 2 = -x + 3 (at right)
   
• EVERY NUMBER IS A SOLUTION
  - the statement is always true,
  for example, 5 = 5 or x = x, or,
  x + 5 = 5 + x
   
• THERE IS NO SOLUTION
  - the statement is false,
  for example, 10 = 5 or 2 + 3 = 6 + 1, or
  7 + x = x + 5.

      Consider the following examples.

3 + 2x = 1       The two expressions "3 + 2x" and "1" do not mean the same thing but there may be a time when they are equal. They are equal when -1 is used as the value of x.       The solution is -1. That is when the lines cross. That is when the expressions are equal. 3 + 2x = -1       The expressions "3 + 2x" is graphed. The expression "-1" is not. Mentally graph "-1" and find all points where they are equal.       They are equal when -2 is used as the value of x. The solution is -2. That is when this lines cross. That is when the expressions are equal.

2x + 3 = 4x - 1       The two expressions "2x + 3" and "4x - 1" don't mean the same thing but there may be a time when they are equal. They are equal, the lines cross, when x is 2, and only when x is 2.       The solution is 2.

x + 2 = -x + 3       The two expressions "x + 2" and "-x+3" do not mean the same thing. There may be a time when they are equal.       To solve x + 2 = -x + 3, find when they are equal. Find where the lines cross. They cross only when x is 1/2. The solution is 1/2.

x + 5 = 5 + x       The two expressions "x + 5" and "5 + x" mean the same thing but are said in different ways. They are always equal no matter what value of x is used in both expressions.       To solve x + 5 = 5 + x, find when they are equal, where the lines cross. They cross for every value of x. They are always equal. The solution is all numbers.

7 + x = x + 5       The two expressions "7 + x" and "x + 5" mean different things -- 7 more than a number, 5 more than a number. They are never equal no matter what value of x is used in both expressions.       To solve 7 + x = x + 5, find when they are equal, where the lines cross. They never cross for any value of x. There is no solution.

 

      The graphic technique of finding where the curves intersect to find the solution works all the time, but, works best when the intersections are easy to find/compute/read.

      More examples follow.

 

 

Solve: |x+3| -2 = |x-6|       The two expressions |x+3| -2 and |x-6| are equal for only 1 value of x.   When x is 3, |x+3| -2 equals |x-6|. The solution to the equation |x+3| -2 = |x-6| is 3.     Solve |x-2| = 1       The two expressions |x-2| and 1 are equal for 2 values of x.   When x is 3 or when x is 1, |x-2| equals 1. The solutions to the equation |x-2| = 1 are 3 and 1.     Solve |x-2| = -1       The two expressions |x-2| and -1 are never equal. It is never the case that the graphs cross. It is never the case that |x-2| equals -1. There is no solutions to the equation |x-2| = -1.     Solve cos( ) = (2)/2, where - < < Solve cos( ) = (2)/2, where 0 < < 2       We'll use the same picture and a bit of arithmetic to solve both equations.       The two expressions cos( ) and (2)/2 are equal for 2 values of in the restricted interval between - and . Graphs of the two expressions cross at two values of . When is - /4 and when is /4, the expressions are equal. These are the solutions.       For the interval from 0 to 2 , there are two solutions to cos( ) = (2)/2. One solution is /4. To get the other solution, take - /4 and add 2 . The result is 7 /4. This is the second solution to the equation cos( ) = (2)/2. This solution is readily visible in the next example.

Solve cos( ) = (2)/2

      The two expressions cos( ) and (2)/2 are equal for an infinite number of values of because the cosine function is periodic (infinitly repeating) and the (2)/2 is constant (always the same). Graphs of the two expressions cross an infinite number of times, each equal to 7 /4 ± 2 n and /4 ± 2 n, where n is 0, 1, 2, 3, ... These are the solutions.

 

 

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