Solving Equations

Example 1:

Substituting $2$ for $x$ in

$3x+5=11$

gives

$3\left(2\right)+5=11$ , which says $6+5=11$ ; that's true!

So $2$ is a solution.

Example 2:

The equation

$x^2=x$

has two solutions, $0$ and $1$ , since

$0^2=0$ and $1^2=1$ . No other number works.

Example 3:

The equation

$x+1=1+x$

is true for all real numbers . It has infinitely many solutions.

Example 4:

The equation

$x+1=x$

is never true for any real number. It has no solutions .

Example 5:

Solve the equation

$x^2=\sqrt{x}$

over the domain $\{0,1,2,3\}$ .

This is a slightly tricky equation; it's not linear and it's not quadratic , so we don't have a good method to solve it. However, since the domain only contains four numbers, we can just use trial and error.

$\begin{array}{l}{0}^2=\sqrt{0}=0\\ 1^2=\sqrt{1}=1\\ 2^2\ne \sqrt{2}\\ 3^2\ne \sqrt{3}\end{array}$

So the solution set over the given domain is $\{0,1\}$ .

Example 6:

The equation

$x=y+1$

is true when $x=3$ and $y=2$ . So, the ordered pair

$(3,2)$

is a solution to the equation.