Algebra

Question 1

Algebraic identities

Answer 1

Question 2

Rules of exponents

Answer 2

Question 3

Rules to produce equivalent equations

Answer 3

Rule 1: When the same constant is added to or subtracted from both sides of an equation, the equality is preserved and the new equation is equivalent to the original equation.

Rule 2: When both sides of an equation are multiplied or divided by the same nonzero constant, the equality is preserved and the new equation is equivalent to the original equation.

Rule 3: When an expression that occurs in an equation is replaced by an equivalent expression, the equality is preserved and the new equation is equivalent to the original equation.

Question 4

rules of solving linear equations

Answer 4

1. eliminate fractions (if any). multiply by the least common denominator to get rid of fractions.
2. expand parenthesis. use identities as distributive properties.
3. move and combine like-terms together.

Question 5

Solving linear equations of two variables (substitution)

Answer 5

In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation.

example:

4 x + 3 y = 13
x + 2 y = 2

Express x in the second equation in terms of y as x = 2 −2𝑦.

Substitute 2 −2𝑦 for x in the first equation to get 4 (2−2𝑦) +3𝑦 =13.

Replace 4 (2−2𝑦) +3𝑦 =13 with (8 - 8y) + 3y = 13

Combine like terms to get 8−5𝑦 =13.

Solving for y gives y = -1.

now:
4 x + 3 (-1) = 13
x + 2 (-1) = 2

in the second equation:
x = 2 - 2 (-1)
x = 4

Question 6

Solving linear equations of two variables (elimination)

Answer 6

In the elimination method, the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other.

example:

4 x + 3 y = 13
x + 2 y = 2

multiply the second equation by 4 so they have the same coeffient of x:

4 x + 3 y = 13
4 x + 8 y = 8

if you substract the equations:

4 x - 4 x
3 y - 8 y
13 - 8
you get -5y = 5
y = -1

so x = 2 - 2 (-1)
x = 4

Question 7

Quadratic formula

Answer 7

A quadratic eqaution: where 𝑎, 𝑏, and 𝑐 are real numbers and a, is not equal to 0𝑎≠0. Quadratic equations have zero, one, or two real solutions.
𝑎𝑥^2+𝑏𝑥+𝑐=0

to solve, you can use the quadratic formula

Question 8

Solving quadratic equations by factoring

Answer 8

Some quadratic equations can be solved more quickly by factoring.

Example :
2𝑥^2−𝑥−6=0
can be factored as (2𝑥+3)(𝑥−2)=0

product is equal to 0, at least one of the factors must be equal to 0, so either 2x + 3 = 0 or 𝑥−2=0.

If 2x + 3 = 0
then 2𝑥=−3
x = -3/2

If 𝑥−2=0, then x = 2

Thus the solutions are negative, −3/2 and 2.

Question 9

solving linear inequalities

Answer 9

To solve an inequality means to find the set of all values of the variable that make the inequality true.

For example, the inequality 4𝑥+1≤ 7 is a linear inequality in one variable, which states that 4𝑥+1 is less than or equal to 7.

The procedure used to solve a linear inequality is to simplify the inequality by isolating the variable on one side of the inequality, using the following two rules.

Rule 1: When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved.

Rule 2: When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative. In either case, the new inequality is equivalent to the original.

example:

4𝑥+1≤ 7

substract 1 from both sides

4𝑥 ≤ 6
𝑥 ≤ 6/ 4
𝑥 ≤ 1.5

4(1.5)+1 =7
so x ≤ 7

Question 10

Functions

Answer 10

A function is like a machine: you put in x, do some math, and get an output.
Example:

f(x)=3x+5, so if
x =2
f(2)=11.

The domain is the set of numbers we can use. Some functions exclude numbers, like when we cannot divide by 0.

The absolute value function gives the distance from zero, so

h(x)=∣x∣
means
h(−3)=3.

Question 11

average (arithmetic mean)

Answer 11

Average=
Numberofvalues / Sumofallvalues
​

Question 12

Distance, rate, time

Answer 12

rate and speed are the same.

Question 13

simple interest

Answer 13

𝑉=𝑃(1+𝑟𝑡/100)

V = final value of the investment
P = initial deposit (principal)
r = annual interest rate (as a percentage)
t = time in years

Question 14

compound interest

Answer 14

Unlike simple interest, compound interest keeps growing because you earn interest on top of your interest.

𝑉=𝑃(1+𝑟/100)^𝑡

Question 15

Compound Interest (Compounded Multiple Times Per Year)

Answer 15

If interest is compounded quarterly, monthly, or daily, we use this formula:

V = P (1+ r /100n) ^nt

Where n is the number of times per year interest is compounded:

Quarterly: n= 4
Monthly: n=12
Daily: n=365