Variables in Mathematics is a symbol generally a letter used to represent an unknown value, or a substitution for a complex and common value in equations. Example: ## Variables Variables can be extremely confusing in algebra due to usage, beginners don't quite get this concept as its completely new and foreign. One can think of variables as a container, box or placeholder containing unknown values. You can represent variables with almost any symbol, the most common are: - `x` - `y` - `z` In other fields greek letters, and other symbols are used to describe complex variables such as speed, force, electrical currents, magnetic force etc... **A practical example trying to explain variables:** A local hardware store pays Jack $80 for a day's worth of work. Jack receives tips while helping customers load equipment and gear onto their trucks. In this example we know that jack makes $\$80 + \text{tips}$. To formulate an expression to determine how much money jack makes we can represent $x = \text{tips}$. $80 + x = ?$ where x is the amount of tips (which in this case can change hence a variable.) Lets assume today Jack made `$10` in tips on top of his salary, how much money has jack made? $x = 10$ $80 + x = ?$ $80 + 5 = 85$ This equation shows that `Jack made a total of $85 today.` ## Coefficients & Constants Coefficients are useful for presenting multiplication between items. The first example in this note showcases it nicely, but lets go a bit more in-depth. #### Coefficients **A practical example trying to explain coefficients:** Its Amanda's birthday. Amanda brings 2 cupcakes, 3 candy bars & 2 fruits for each child in the classroom, If there's 30 children in the classroom how many items does Amanda need to carry at once? $\text{all the children} = x$ $\text{total items} = y$ $y = x(2\cdot3\cdot2)$ $\text{let } x = 30$ $y = 30(2\cdot3\cdot2)$ $y = 30(12)$ $y = 360$ $\text{Amanda will bring 360 items to school for her classmates on her birthday.}$ #### Constants Constants are fixed values that never change in an expression or equation. Unlike variables, which can represent different values, constants always stay the same. **A practical example to explain constants:** Maria works at a pizza shop that charges a $3 delivery fee for every order, regardless of the pizza size or toppings. On top of this fixed fee, customers pay $12-15 for each pizza they order, this example considers only $12 pizzas In this scenario: - The $3 delivery fee is a **constant** (it never changes) - The number of pizzas ordered is a **variable** (it can change) - The $12 per pizza is a **coefficient** (it multiplies the variable) $\text{Total cost} = y\cdot x + 3$ Where: - $y = \text{cost per pizza for this customer}$ - $x = \text{number of pizzas}$ - $12= \text{coefficient}$ - $3 = constant$ **Example calculations:** - If someone orders 2 pizzas: $12(2)+3=24+3=\$27$ - If someone orders 5 pizzas: $12(5)+3=60+3=\$63$ - If someone orders 2x 15$ pizzas: $15(2)+3=30+3=\$33$ Notice how the $3 delivery fee (constant) stays the same in every calculation, while the pizza cost changes based on the number ordered and pizza cost. In more complex formulas constants can be presented as variables which makes it easier to memorize formulas. Here is a very simple formula from physics: $F = G\frac{M_1 \cdot M_2}{r^2}$ If this formula was to expand it would look something like this, hence describing constants with vars are helpful. $F = 6.674 \times 10^{-11} \frac{M_1 \cdot M_2}{r^2}$ ## Terms Terms in mathematics are separate units of an expression. A single term can contain multiple multiplication / division. $2x^2 + 9y - 4x$ → This expression contains 3 terms denoted below: $2x^2$, $9y$ & $-4x$ In an equation terms are separated by addition or subtraction symbols. Terms can also be nested by using brackets / parenthesis **[[PEDMAS]]**. $5(x + 1) - 3y^2$ → This expression contains 2 terms, and 1 nested term: $5(x + 1)$ & $-3y^2$ As shown in the examples above we have the subtraction symbol in both $-4x$ & $-3y^2$. The subtraction symbol belongs to a term. You could rewrite both equations in this form if it makes more sense: $2x^2 + 9y + (-4x)$ $5(x + 1) + (-3y^2)$ As shown in the examples above, we haven't really showcased how fractions could play a role in terms. $\frac{1x}{2}\cdot y - 12^2z$ → This expression contains 2 terms $\frac{1x}{2}\cdot y$ & $- 12^2z$ Terms are only split by $+$ or $-$ signs, Any multiplication done to variables / numbers are considered to be the same term. ## Algebraic Expressions In previous notes you can see references to _"Expressions"_. One/Multiple terms combined together create **algebraic expressions** when variables are present. Here is a quick list of **Algebraic Expressions** - $y\cdot x + 3$ - $2x^2 + 9y -4x$ - $5(x + 1) -3y^2$ - $G\frac{M_1 \cdot M_2}{r^2}$ - $\frac{x}{2}\cdot y - 12^2z$ An algebraic expression requires at least one term (which includes a variable). Quick list of **single term** Algebraic Expressions: - $\frac{x}{y}$ - $2x^y$ - $G\frac{M_1 \cdot M_2}{r^2}$ Quick list of **non-Algebraic Expressions**: - These expressions are all considered **numeric / arithmetic expressions** - $7 \times 91$ - $5(3 + 2)^2 - 1$ - $6.674 \times 10^{-11}$ Algebraic Expressions, unlike Arithmetic Expressions can't be solved without knowing the values of the variables present in an equation. $y\cdot5 + 1$ $\text{let } y = 5$ → We're given the value of $y$ so the expression can be evaluated. $5\cdot5 + 1= 26$ ## Types of Expressions #### Based on Variables - **Algebraic Expressions**: contain variables (e.g., $2x + 5$, $x^2 - 3y$) - **Arithmetic/Number Expressions**: contain only numbers (e.g. $5 + 3 \times 2$) #### Number of Terms - **Monomial** - single term (e.g., $5x^2$,$−7y$, $\frac{x}{3}$) - **Binomial** - two terms (e.g., $3x + 2$, $x^2 - 5$) - **Trinomial** - three terms (e.g., $x^2 + 3x - 4$, $2a + b - 7$) - **Polynomial** - general term for multi-term expressions, binomial + terms #### Based on Highest Power - **Linear** - degree 1 (e.g., $2x + 3$, $5y−1$) - **Quadratic** - degree 2 (e.g., $x^2 + 2x + 1$, $3y^2 - 4$) - **Cubic** - degree 3 (e.g., $x^3 + 2x^2 - x + 5$) ## Like Terms Like terms can be combined, like terms have the same variables raised to the exact same powers. These terms can be combined by adding coefficients while keeping the same variable(s) and exponent(s). Examples of **Like Terms**: - $x + 4x$ → both have the same variables raised to $x^1$. $= 5x$ - $-4 + 11$ → constants are also considered like terms. $= 7$ - $7x + 4y$ → different variables not considered like terms $= 7x + 4y$ - $8y^2 + y\cdot y$ → considered like terms $y \times y$ = $y^2$. $= 9y^2$ - $9x^3y^2 - x^3y^2$ → considered like terms. $= 8x^3y^2$ Like terms can be combined to make equations / expressions more concise. ## Simplifying Expressions Usually when solving complex equations you can simplify the expressions on both sides of the equality / inequality signs. Simplifying can help making it more approachable, by allowing you to move dense terms around. There are several different ways to **Simplify Expressions**: - Combining Like Terms → Explained in the previous section - Distributive Property → Multiplication / division between terms - Removing Brackets / Parens → Reduces the amount of nesting - Combining Constants → Reduces the amount of terms in the expression #### Combining Like Terms Explained [[#Like Terms|Here]]. #### Combining Constants Combining constants are just combining any number values possible to reduce the amount of terms. $3y - 1 + 5x + 7 = 5x + 3y + 6$ ← In this case we combined $-1$ and $7$ which $= 6$ #### Distributive Property This section will just quickly cover on how the distributive property works. Here is a [[Learnings/Topics/Mathematics/Algebra 1/Distributive Property|full note]] specifically on all the nuances for the distributive property. The distributive property allows you to simplify a tererm that allows you to multiply the terms outside of brackets / parens. **Example:** $4x(5 + 9) = 20x + 36x$ In the example above we did 2 separate multiplications for the different terms inside of the brackets. $4x \times 5$ and $4x \times 9$. These terms are then joined with the exact same symbols they had internally $-$ In this case it was a plus symbol $+$. All the multiplication / division properties are applied here, like sign negation. $-1(5x + 9) = -5x - 9$ ← In this case we did a sign negation. #### Removing Brackets / Parens If an equation has unnecessary brackets / parens you can remove them. As a friendly reminder make sure the expression stays the same!! $5+(3x−2)=5+3x−2=3x+3$ ← We removed the unnecessary grouping symbols and combined contstants. ## Evaluating Expressions Evaluating expressions is the process of replacing one or more variables in an expression based on the given variable values to get an answer for that expr. $\text{Evaluate where let } x = 5$ 1 . $x + 22 = 5 + 22 = 27$ 2 . $8x = 8\cdot 5 = 40$ $\text{Evaluate where let x } = 10 \text{ and let y } = 9$ 1 . $7x - y = 7 \cdot 10 - 9 = 63 - 9 = 53$ 2 . $x + \frac{18}{y} = 10 + \frac{18}{9} = 10 + 2 = 12$ As you can see we just replaced the variables with their actual values. As a further improvement on this, always substitute the variables in with parens. **You need to keep the expressions INTEGRITY. Here is an example:** $x^2 - 3x + 5 \text{ where let }x = 4$ Wrong: $-4^2 - 3(- 4) + 5$ $-16 + 12 + 5$ $= 1$ Correct: $(-4)^2 - 3(- 4) + 5$ $16 + 12 + 5$ $= 33$ In the above example "wrong" example,the equation states "negative of 4 squared". Which evaluates to $-(4 \cdot 4) = -16$ In the correct equation we're saying: negative 4, squared $-$ in n he correct example we're referring to squaring the term, aka $(-4) \cdot (-4) = 16$ > [!ERROR] > Always use parenthesis for substituting variables, you need to preserve the original meaning of the expression! ## Practice Tests - [[Equation+Definition+Bonus+Practice+Test.pdf]] - [[Equation+Definition.pdf]] - [[Variables+and+Expressions+Bonus+Practice+Test+Algebra+1.pdf]] - [[Variables+and+Expressions+Practice+Test.pdf]]