Some systems of equations have solutions that are not whole numbers. When this happens, it is often preferable to use fractions instead of decimal approximations because fractions represent exact values.

 

Example 3

Solve the following system of equations. Verify the solution.

If you used the decimal approximations «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»269«/mn»«mn»115«/mn»«/mfrac»«mo»§#8773;«/mo»«mo»-«/mo»«mn»2«/mn»«mo».«/mo»«mn»34«/mn»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»279«/mn»«mn»115«/mn»«/mfrac»«mo»§#8773;«/mo»«mo»-«/mo»«mn»2«/mn»«mo».«/mo»«mn»43«/mn»«/math», the verification would be

Left Side
 Right Side
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»13«/mn»«mi»x«/mi»«mo»-«/mo»«mn»8«/mn»«mi»y«/mi»«mo»+«/mo»«mn»11«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»13«/mn»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo».«/mo»«mn»34«/mn»«mo»)«/mo»«mo»-«/mo»«mn»8«/mn»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo».«/mo»«mn»43«/mn»«mo»)«/mo»«mo»+«/mo»«mn»11«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«mo».«/mo»«mn»02«/mn»«/math»
 «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»L«/mi»«mi»S«/mi»«mo»§#8773;«/mo»«mi»R«/mi»«mi»S«/mi»«/math»

Left Side
 Right Side
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»7«/mn»«mi»y«/mi»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»7«/mn»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo».«/mo»«mn»34«/mn»«mo»)«/mo»«mo»+«/mo»«mn»7«/mn»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo».«/mo»«mn»43«/mn»«mo»)«/mo»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»24«/mn»«mo».«/mo»«mn»03«/mn»«/math»
 «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»24«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»L«/mi»«mi»S«/mi»«mo»§#8773;«/mo»«mi»R«/mi»«mi»S«/mi»«/math»

The two sides are approximately equal, so it can be assumed the difference is due to rounding. This method is acceptable, but provides less confidence than a verification using the exact values.

Verify the solution.