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.
|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?"||
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.
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.
Note how the problems at the top of this page are solved mentally by thinking about what the equation tells us. Do this for the following problems to find the solution to each in your head.
Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.