• pre-algebra
• graphs and lines
• functions
• equations and inequalities
• complex numbers
• polynomial
• logarithm
• angle functions
• systems of equations
• matrix algebra
• counting and probability
• conic sections

Try our resources for yourself, and you find that getting better grades in less time isn't that hard at all! 

Dear Professor Cram:

What is the solution to the system of equations: x = y and x = -y?

KG, US Navy

KG, thank you for this algebra question.

The easiest way to solve this system of equations problem is by using the graphing method.

(Use the illustration to the left for a helpful guide.) If we graph the first equation, we get a diagonal line passing through the origin sloping upwards. (See the blue line in the illustration.) Graphing the second equation gives us a diagonal passing through the origin and sloping downwards. (See the pink line in the illustration.)

From visual inspection, we can see that the lines intersect at one place only -- the origin. Thus, the solution to this system is (0,0).

Dear Professor Cram:

Jerry was 1/3 as young as his grandfather 15 years ago. If the sum of their ages is 110, how old is Jerry's grandfather? What is the easiest way to calculate this? Thanks

Kaye B, Riverdale

Thanks for your word-problem question, Kaye. This is a system of linear equations problem, which we'll solve using the Substitution Method. (Learn more about the Systems of Linear Equations Substitution Method.) Let's review the facts first:

• Jerry's age today is J
• Grandpa's age today is G
• Jerry's age 15 years ago is (J-15)
• Grandpa's age 15 years ago is (G-15)

In equation form, then, we can express their ages 15 years ago and today like so:

• #1: J - 15 = (G-15)/3 (age 15 years ago)
• #2: J + G = 110 (age today)

Solve equation #2 for grandpa's age by subtracting J from both sides, yielding:

#2: G = 110 - J

Now substitute this definition of G into equation #1 so we can solve for J:

#1: J-15 = ((110-J)-15)/3, or
#1: J-15 = (95-J)/3, or
#1: 3(J-15) = (95-J), or
#1: 3J-45 = 95-J, or
#1: 3J = 95-J+45, or
#1: 3J+J = 95+45, or
#1: 4J = 140, or
#1: J = 35

So if J=35, then that means Jerry's age today is 35 years old. Using equation #2, that means Grandpa's age today is 110-35, or 75 years old.

The final step in solving this problem is one that many students forget: checking your answer! Let's put our values for J and G into equations #1 and #2 and make sure they work:

#1: 35-15 = (75-15)/3, or
#1: 20 = (60)/3, or
#1: 20 = 20 CORRECT

#2: 75 = 110-35, or
#2: 75 = 75 CORRECT

The correct answer, then, is that Jerry's grandfather is 75 years old.

Look at a typical algebra textbook. It gives one or two examples, and tells you what the answers are, but it doesn't really explain how you get them. It's like when I was a kid learning how to tie my shoes with the other kids. The adult told us to cross the laces, twist one underneath and pull, make a loop and hold it, then take the other lace and. do something. (This part I had a really hard time getting. Showing me once or twice definitely didn't help, nor did showing me shoes that were perfectly tied.) Eventually I got it, but that was after trying over and over again and asking if I'd gotten it right yet.

The same process holds true for factoring trinomials. There are guidelines for doing it, but no hard-and-fast steps to follow every time. The only way to learn it properly is to practice repeatedly, referring back to the guidelines, until you get it.

The Factoring Trinomials Cramlet™ has several features that make it effective where textbooks are not.

• The See Tutorial button gives a good overview of the techniques you can use to factor trinomials, walking you through an example complete with typical missteps until you find the correct answer. This tutorial is available at any time while you're running the Cramlet™.
• The New Problem button instantly gives you another problem to solve. Unlike some programs that use random number generation, though, all these problems actually have a solution so you're not wasting your time.
• This Cramlet™ also sports a Show Hints button, available at any time in the program, which displays the guidelines covered in the tutorial. These reminders are very helpful if you get stuck.
• Check my Answer takes your guesses, compares them with the correct answer, and tells which coefficients and signs you got right. This is really useful where you might pick the right numbers, but still get the wrong answer because you put the minus sign in the wrong place. You can keep trying and clicking this button until you get the problem right.

Factoring Trinomials is one of our most popular Cramlets™, and countless students have used it to master this concept. Of course, its best feature is that, unlike a textbook, it is available for free to every student who signs up for a free registration.