Algebra

Question 1

What does it mean when an equation is true?

Answer 1

When the statement made by the equation is factually correct. E.g the equation “x + 0 = 3” can be both true or false. However when we replace the variable x with a 3 then the equation is true. Replacing x with any other number than 3 will make the equation false.

Question 2

What is the solution to the following equation?
5x - 2y = 3
-3x + 3y = 0

Answer 2

X = 1
Y = 1

Question 3

What is a single variable equation?

Answer 3

An equation which requires to solve for only one variable. “y + 3 = 8” is a single variable equation because if we solve for y we have a true equation.

Question 4

You can add and multiply both sides of an equation by any non-zero real number without changing the truth of that equation.

True or false?

Answer 4

True

Question 5

What is a linear equation?

Answer 5

An equation made up of a sum of terms, each of which consists of a constant real number multiplied by a variable that is raised to the first power. If a variable in the equation is raised to the power higher than 1 then the equation is non-linear.

Question 6

What is a variable?

Answer 6

A variable is a numerical value that is not fixed. As the name implies, the value of a variable varies. Variables are used to form equations. E.g. “x = 5” the variable x is equal to 5

Question 7

What does it mean to “collect like terms” while simplifying equations?

Answer 7

When we look for repeated additions and turn them into a multiplication. E.g. “3 + 3” is a repeated addition, we turn it into a multiplication “23”. Same goes with “a + a” we make it “2a”.

Question 8

Simplify the following equation by finding “like” terms:

3xy + 2xy + 5 = z

Answer 8

5xy + 5 = z

Question 9

What does “factoring an equation” mean?

Answer 9

Factoring an equation involves writing some expression as the product of one or more factors which themselves are sums.

Question 10

Using factoring, simplify the following equation:

x + yx = 3

Answer 10

We are essentially adding x to itself y number of times. So if we are simply then adding x to the result of the expression we are adding x to itself y + 1 number of times. So the answer is “(1 + y)* x = 3”

Question 11

Using factoring, simplify the following equation:

5x + xz^2 = 7

Answer 11

(5 + z^2) * x

Question 12

What does it mean to “expand” an equation?

Answer 12

Expanding an equation involves using the distributive property of multiplications to break down (enlarge) the equation. This is done in some cases so that we can then find like terms and finally factor the equation.

Question 13

Expand the following equation:

x * (y + z + a) = 7

Answer 13

xy + xz + xa = 7

Question 14

Simplify the following expression by expanding, finding like terms and factoring.

x (a + y + c) + y (x + a + b) + z (x + y)

Answer 14

Expand: xa + xy + xc + ya + yb + zx + zy
Find like terms : xa + 2xy + xc + ya + yb + zx + zy
Factor: x * (2y + c + z) + y * (a + b + z)

Question 15

What does the FOIL method stand for?

Answer 15

First, outside, inside, last.

Question 16

Apply FOIL to the following equation:

a + b) (c + d

Answer 16

ac + ad + bc + bd

Question 17

What is a function?

Answer 17

A set of rules that associates every element of a set A (aka “domain”)(a set is a unordered collection of items e.g. Real numbers are a set) with exactly one element of a set B (called the range). Typically, both of these sets will contain a range (possibly infinite) of real numbers.

Question 18

What is the purpose of inverse functions?

Answer 18

To undo the operation a function performed.

Question 19

How is an inverse function denoted as opposed to a function?

Answer 19

Function: f(x) = 2x
Inverse: f-1(x) = 1/2 x

Question 20

How do we convert a function to an inverse function?

Answer 20

We treat f(x) as a variable and solve for the equation as we would any other equation.