System of equations 

A set of one or more equations involving a number of variables. An example would be 2y + 3x = 38 and y – 2x = 12.

The solution to the equation is x = 2, y = 16. This is because x = 2 and y = 6 works with each equation in the system.

Example 1 

y = 3x – 7

y = -x + 9

Use algebra to find the variables’ values:

x = 4, y = 5

To verify the answer, plug in the numbers of the variables.

5 = 3(4) – 7

5 = -4 + 9

Example 2 

y = x – 2

y = 3/4x – 4

Use algebra to find the variables’ values:

x = -8, y = -10 

Verify the answers:

-10 = -8 – 2

-10 = 3/4(-8) – 4

Why

By knowing how to solve systems of equations you know how to solve applications where there are unknown variables. It is useful in math, especially with solving word problems involving finding a missing variable.

Vocabulary

Variable- Symbols that represent different values in various situations

Equation- A mathematical statement that shows that two mathematical expressions are equal.

Linear patterns

5, 8, 11, 14, 17

Starts at 2 and increases by 3.

50, 40, 30, 20 ,10

Starts at 50 and decreased by 10.

When we have a linear number pattern we can use the information given to create an equation. In the example format of: y=3x+2.

Examples

5, 8, 11, 14, 17

In this example, the equation is y=3x +2. You can create a T-Chart to help you find the equation. We can see that 3 times x equals 3, and plus 2 equals 5. This works with all of the numbers in the pattern as well. 2 x 3 = 6 —> 6+2=8, and so on.

50, 40, 30, 20, 10

In this example, the equation is y= -10x+60. We can see that -10 times x equals -10 and plus 60 equals 50. This equation works to get the rest of the numbers as well.

Why?

It is important to know how to collect information from a linear pattern and be able to create an equation with it because in math you need to understand how the equations work with x and y to easily work with linear relations, t-charts, and graphs.

Vocabulary 

Linear pattern- A pattern that forms a straight line. A linear number pattern is a sequence of numbers where the difference between every term is the same.

The degree of a polynomial is given by the term or monomial with the highest degree.

A constant Polynomial: has a degree of 0

A linear Polynomial: has a degree of 1

A Quadric polynomial: has a degree of 2

A Cubic Polynomial: has a degree of 3

A Quartic Polynomial: has a degree of 4

A Quintic Polynomial: has a degree of 5

Examples

| This is a Quadratic Polynomial because it has a degree of 2.

| This is a cubic polynomial because it has a degree of 3.

x+5           | This is a linear polynomial because it has a degree of 1.

Why

By knowing how to recognize a polynomial by the degree it will make classifying polynomials easier and will save time when you are asked to classify a polynomial in future math situations.

Vocabulary 

degree- The degree is the highest exponent value of the variables in the polynomial.

Factoring + bx + c

To factor this trinomial, you need to find two integers that have a product equal to c and a sum equal to b. If there are no two integers that work, then the polynomial cannot be factored.

Examples

+ 11x +10

Step 1: Find two numbers that multiply to get 10 and add to get 11.

Step 2: Once you have found the two numbers that work, write out the polynomial like this:

(x +10)(x + 1)

Step 3: This shows the factored version of the trinomial + 11x +10

+ 6x + 4

Step 1: Find the GCF that all three numbers have in common.

Step 2: Create a box visual to help show your work in factoring the polynomial.

Step 3: Write out the factored polynomial and find two numbers that have a have a product equal to 2 and a sum equal to 3.

2(x+2)(x+1)

Why?

To know how to factor polynomials by inspection is a great skill to have because it can help solve math problems that include polynomials in the future and will make the process easier and quicker.

Vocabulary

Polynomial- A polynomial is an expression that is composed of variables, constants, and exponents, that are combined using mathematical operations including: addition, subtraction, multiplication and division.

Trinomial- A polynomial consisting of three terms.

GCF- Greatest common factor

Multiplying two binomials: distributive property

There are many different quick and efficient ways to multiply two binomials. One of them being distributive property.

Examples

(x-6)(x+12)

Step 1: Multiply the x in the first bracket with the x in the second bracket.

Step 2: Multiply the x with the 12.

Step 3: Multiply the -6 with the x in the second bracket.

Step 4: Multiply the -6 with 12.

Step 5: Write the products of the numbers in a line. This is the expanded form.

Step 6: Simplify. Add the like numbers together.

Multiplying two Binomials: Area Model

Using an area model to multiply two binomials is an easy and a great visual way to do so.

(2x+3)(x+12)

Step 1: Create a rectangular diagram and write out the monomials, one on top and one on the side.

Step 2: In the first box you put the product of x multiplied by 2x.

Step 3: In the second box you put the product of x multiplied by 3.

Step 4: Do the same with the bottom two boxes. The bottom left box is the product of 12 and 2x, the right box is the product of 12 and 3.

Step 5: Write the numbers in the boxes in a line. This is expanded form.

Step 6: Simplify.

a binomial is a polynomial that is the sum of two terms,

Why?

It is important to know the different ways in making multiplying binomials easy and quick. It will help in polynomial related math problems in the future and make it much easier for you to do the steps in solving the problems.

Vocabulary

Binomial- A polynomial that consists of two terms.

Trigonometry

The study of triangles and the relationships between sides and angles in a triangle.

Names of sides

Hypotenuse- The hypotenuse of a right triangle is always the side opposite of the right angle. It is the longest side in a right triangle.

Opposite- The opposite side is the side across from a given angle.

(The given angle is c)

Adjacent- The adjacent side is the non-hypotenuse side that is next to a given angle.

Why?

Knowing the names of a right triangle is important because it is crucial in determining the ratios of a triangle. It will also help when you’re using pythagorean theorem to solve math problems, if you know the sides of a right triangle it will be easier to identify the hypotenuse and make the process quicker in solving trigonometry equations.

Vocabulary

Theta- A variable to represent a measured angle

Right triangle- A triangle with one right angle, that is 90°

Exponent Laws

Laws that are used for simplifying expressions with exponents. These laws are helpful in simplifying expressions that have decimals, fractions, irrational numbers, and negative integers as their exponents. It helps makes the process much easier.

Multiplication Law– When multiplying like bases, keep the base the same and add the exponents.

Division Law– When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.

Power Law– To raise a number with an exponent to a power, you multiply the exponent and the power.

Zero Exponent Law- Any number raised to the power of 0 is equal to 1.

Why?

Exponent laws are important to know so that you can easily solve many mathematical problems involving repeated multiplication. The exponent laws simplifies the divisions and multiplication operations to help solve the problems and simplifies the calculations.

Vocabulary

Exponents– A short way to write repeated multiplication, written in exponential form such as

Powers– A power is a number written in exponential form. Consisting of a base and an exponent.