Solving Equations

Solving in your head

When we first look at an equation, we see if we can guess the answer by mentally trying numbers and thinking about what the equation is asking us.

Examples
Note how the translation of the equation in the left column leads to the solution in the right column.

to solve		we think		to find
		"I'm looking for a number whose square root is 4"
		"What number for x gives 2x = 3?"
		"Three divided by what number gives -1?"
		"What exponent will make the exponential be 1?"
		"What numbers make each factor zero?"
		"What numbers make the numerator zero?"
		"What values make the sine value equal to 1?"

Operations on equations

If we can't solve an equation mentally, we may be able to solve it by applying operations to it: the rules of equality allow us to perform the same operation on both sides of an equation. Thus, we can add or subtract the same number from both sides, or we can multiply both sides by the same number (as long as we avoid dividing by zero), etc. This is how we solve equations in the following topics.

Solving exactly versus solving approximately

Some equations we can solve exactly, often by using algebra. For example, the equation 7x - 1 = 0 has the solution x = 1/7. Other equations may be more difficult or even impossible to solve exactly. In these cases it may be possible, and easier, to obtain an approximate solution by using a graph or a numerical method on a calculator. The equation 7x - 1 = 0 has the approximate solution , since (1/7) = 0.142857...). In this case we use the approximately sign () to emphasize that the solution is not exact.

Examples
For example, has the exact solutions which lead to the approximate solutions or or (since ). Notice that there are only two exact solutions to this equation, but many possible approximate solutions depending on how much accuracy is required.