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### Course: Algebra 2>Unit 10

Lesson 6: Solving equations by graphing

# Solving equations by graphing: intro

Some equations are hard to solve exactly with algebraic tools. We can always solve an equation by graphing it, although the solution might not be exact. This is an example of how to solve a relatively simple equation graphically.

## Want to join the conversation?

• When are we allowed to do this? And wouldn't the first example make it a non-invertible function?
• I am pretty sure you are allowed to do this with any equation. I may be wrong, but that is what I think.
• For example 1, which is y=(3/2)^x=5, couldn't you also find the point of intersection with logarithms? (3/2)^x=5 is an exponential expression. Therefore, it could also be written as log_3/2(5)=x. Using the change of base formula, the logarithm could be rewritten as log(5)/log(3/2). When I plug this into my calculator, I get x=3.969362296. I rounded up to 4 to get the answer. I know this example was used to show the usefulness of graphing when approximating an answer; however, I believe this example would also be an opportune time to utilize logarithms, especially if graph paper is not available. If you see any errors in my work or logic, please let me know.
• How do you solve this without using graphs or computers?
(1 vote)
• There are generally multiple ways to solve such problems and the possibilities depend on the particular problem.

For the first problem, (3/2)^x = 5, for example, you could find an upper and lower bound for the value of x and then keep shrinking the range of values to get better approximations for x. Alternatively, a more complex solution would involve using some algebra and something called a Maclaurin series (a topic covered in calculus) to directly compute the value for x.

For the second problem, y = x^3 - 2x^2 - x + 1, you could factor the polynomial and see how many distinct solutions there are for each question. You could also compute the derivative (another calculus topic), find the regions over which the function increases/decreases, and check whether each region goes through the particular y-value of each question.

These are only a couple of ways I thought to solve them off the top of my head. There are no doubt other ways to solve them as well.