Solving equations

Summary

Numerical and analytical solutions

• Numerical solutions to an equation: when you find solutions by trying different values for the variable/variables, usually with a computer.
• Analytical solutions to an equation: when you find solutions by successively replacing your equation with a simpler equivalent equation until the solutions appear.
• Most equations cannot be solved analytically. Important exceptions: linear equations, linear systems of equations and quadratic equations.

Methods for creating equivalent equations

• Adding or subtracting the same quantity to both sides of an equation.
• Example: \(x-2=5⟺x-2+2=5+2⟺x=7\) .
• Multiplying or dividing both sides of an equation by a non-zero constant.
• Example: \(2x=6⟺2x/2=6/2⟺x=3\) .
• Care must be taken when we multiply or divide an expression by a variable so that we avoid multiplying/dividing by zero.
• Applying an identity to transform one side of the equation.
• Example: \(2(x+1/2)=5⟺2x+1=5⟺x=2\) by the distributive law.
• Use \(a=b⟹a^2=b^2\) and square both sides of your equation (this may introduce “ fake solution ” )
• Example: \(\sqrt{x}=2⟹x=4\) .
• Use \(a=b⟺log a=log b\) or \(a=b⟺ln a=ln b\) to log both sides of an equation
• Example: \(e^x=6⟺ln e^x=ln 6⟺x=ln 6\) .
• Use \(a=b⟺{10}^a={10}^b\) or \(a=b⟺e^a=e^b\)
• Example: \(ln x=4⟺e^{ln x}=e^4⟺x=e^4\) .
• Use " \(ab=0\) if and only if \(a=0\) or \(b=0\) ".
• Example: \(x(x+1)=0\) then either \(x=0\) or \(x+1=0\) and the solutions are \(x=0, x=-1\) .