Solving Single Variable Equations

These questions typically appear early in the math sections, and only need the 4 basic operations (add, subtract, multiply, divide) to solve them.  

Examples:

$$x-7=12$$

$$4x+3=15$$

$$\frac x6+\frac23=4$$

Per usual on the SAT, they will sometimes try and make these look harder by introducing another variable or constant.  Don’t be fooled – you’re just solving for x.  They key for these is any time you are given an additional value, immediately plug it in.

Examples:

$$\sqrt{x+a}-3=1$$
If a = 1, what is the solution set of the equation above?

If \(x=\frac34y\) and \(y=16\), what is the value of \(3x+2\)?

Strategy:

  1. Perform your order of operations correctly.  You may have learned the acronym PEMDAS for order of operations (if you didn’t, now is a good time!).  Remember, it stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
  2. If you have a square root in the original problem, ALWAYS test for extraneous solutions.  95% of the time you will have one.  If you have a fraction in the original equation, always check that you are not dividing by 0.

Manipulating Polynomials

The last type of “basic” algebra question you will see requires you to know how to manipulate polynomials.  These are the same rules as the single-value equations, but generally require you to deal with exponents in some manner.

A common question:

Which expression is equivalent to $$\left(3x^3-5\right)-3(x^3-x^2-2)$$

A) \(6x^3+3x^2-7\)
B) \(-x^2-7\)
C) \(3x^2+1\)
D) \(3x^3-3\)

Strategy:

  1. Perform your order of operations correctly.  
  2. Combine like terms (same variable and exponent)
  3. Be very careful distributing negatives!

In our example on the previous page, they are testing three skills – do you know your order of operations, do you know how to combine like terms, and do you know how to distribute both multiplication AND a +/- sign.

Here, we will first need to distribute our -3.  Then, we will need to combine like terms (addition and subtraction fall last).

Doing so will bring us to answer C).

Another example the SAT regards as very difficult:

Which of the following is equivalent to \(\left(a+\frac b4\right)^2\)?

A) \(a^2+\frac{b^2}4\)

B) \(a^2+\frac{b^2}{16}\)

C) \(a^2+\frac{ab}4+\frac{b^2}4\)

D) \(a^2+\frac{ab}2+\frac{b^2}{16}\)

When presented with something like this, DO NOT just distribute the exponent!  Notice the answer choices are tempting you to do just that.  Re-write the problem like:

$$\left(a+\frac b4\right)\left(a+\frac b4\right)$$

and then FOIL to get:

$$a^2+\frac{ab}4+\frac{ab}4+\frac{b^2}{16}$$

we then combine our like terms, reduce the middle fraction, and are left with answer D).

Traps

Students miss these problems all the time due to the reasons listed above:  not checking for extraneous solutions, not applying exponent rules correctly, or not keeping close enough track of +/- signs.  If you follow all the algebra rules, these problems are very straightforward.