🔟Elementary Algebra Unit 1 – Foundations

Key Concepts and Definitions

• Algebra involves using letters (variables) to represent unknown quantities and creating equations to solve problems
• Variables are symbols (usually letters) that represent unknown values in an equation
• Coefficients are numbers that multiply a variable in an algebraic expression
• Constants are fixed values in an algebraic expression that do not change
• Like terms have the same variables raised to the same powers and can be combined by adding or subtracting their coefficients
• Simplifying expressions means combining like terms and following the order of operations (PEMDAS) to reduce the expression to its simplest form
• Equations are mathematical statements showing that two expressions are equal, with an equal sign (=) between them
• Solving an equation means finding the value of the variable that makes the equation true

Number Systems and Properties

• The real number system includes all rational and irrational numbers, which can be represented on a number line
  • Rational numbers can be expressed as fractions or terminating/repeating decimals (1/2, 0.5, 0.333...)
  • Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimals (π, √2)
• The commutative property states that the order of addition or multiplication does not change the result: $a + b = b + a$ and $ab = ba$
• The associative property states that grouping does not affect the result of addition or multiplication: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$
• The distributive property allows multiplying a sum by a number: $a(b + c) = ab + ac$
• The additive identity is 0 because adding 0 to any number does not change its value: $a + 0 = a$
• The multiplicative identity is 1 because multiplying any number by 1 does not change its value: $a \times 1 = a$

Basic Algebraic Expressions

• Algebraic expressions consist of variables, coefficients, constants, and operations (addition, subtraction, multiplication, division, exponents)
• Evaluating expressions means substituting known values for variables and simplifying the result
• The order of operations (PEMDAS) is used to simplify expressions: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Combining like terms is an essential skill in simplifying expressions
  • Example: $3x + 2y - x + 4y = 2x + 6y$
• Distributive property is often used to expand or factor expressions
  • Example: $2(3x + 4) = 6x + 8$
• Exponents represent repeated multiplication of a base number
  • Example: $3^2 = 3 \times 3 = 9$

Solving Simple Equations

• The goal of solving an equation is to isolate the variable on one side of the equal sign
• Inverse operations are used to undo operations and maintain equality
  • Addition and subtraction are inverse operations
  • Multiplication and division are inverse operations
• When solving an equation, perform the same operation on both sides to keep the equation balanced
• Steps for solving one-step equations:
  1. Identify the operation being performed on the variable
  2. Apply the inverse operation to both sides of the equation
  3. Simplify and solve for the variable
• Steps for solving two-step equations:
  1. Simplify both sides of the equation by combining like terms
  2. Isolate the variable term using inverse operations
  3. Isolate the variable by dividing both sides by the coefficient

Graphing Fundamentals

• The Cartesian coordinate system is used to plot points and graphs on a 2D plane
• The coordinate plane consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis
• The origin is the point where the x-axis and y-axis intersect, with coordinates (0, 0)
• Points are represented by ordered pairs (x, y), where x is the horizontal coordinate and y is the vertical coordinate
• Quadrants are the four regions formed by the x-axis and y-axis
  • Quadrant I: (+, +), Quadrant II: (-, +), Quadrant III: (-, -), Quadrant IV: (+, -)
• Plotting points involves locating the x-coordinate on the x-axis and the y-coordinate on the y-axis, then finding their intersection
• Linear equations can be graphed by plotting points or using the slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept

Word Problems and Applications

• Word problems present real-life scenarios that require algebraic thinking to solve
• Steps for solving word problems:
  1. Read the problem carefully and identify the unknown quantity (variable)
  2. Assign a variable to the unknown quantity
  3. Write an equation that represents the relationship between the known and unknown quantities
  4. Solve the equation for the variable
  5. Interpret the solution in the context of the problem
• Common types of word problems in algebra include:
  • Distance, rate, and time problems
  • Age problems
  • Mixture problems
  • Work problems
• Translating words into algebraic expressions is a crucial skill for solving word problems
  • Example: "5 less than twice a number" can be written as $2x - 5$

Common Mistakes and How to Avoid Them

• Forgetting to distribute the negative sign when multiplying or dividing both sides of an equation by a negative number
  • Remember to change the sign of each term on the other side of the equation
• Misusing the order of operations (PEMDAS)
  • Always perform operations in the correct order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Confusing the signs when moving terms from one side of an equation to the other
  • When moving a term to the other side of an equation, change its sign (+ becomes -, - becomes +)
• Failing to maintain the equality of an equation when performing operations
  • Always perform the same operation on both sides of an equation to keep it balanced
• Misinterpreting word problems or using incorrect translations
  • Read word problems carefully and ensure that the algebraic expressions accurately represent the given information

Practice Problems and Solutions

1. Simplify the expression: $2x + 3(4x - 1) - 5x$

   • Solution: $2x + 12x - 3 - 5x = 9x - 3$

2. Solve for $x$: $4(2x - 3) = 20$

   • Solution:
     • $8x - 12 = 20$
     • $8x = 32$
     • $x = 4$

3. Graph the equation: $y = -2x + 3$

   • Solution:
     • Find the y-intercept by setting $x = 0$: $y = -2(0) + 3 = 3$, so the y-intercept is $(0, 3)$
     • Find another point by choosing an x-value, like $x = 1$: $y = -2(1) + 3 = 1$, so the point is $(1, 1)$
     • Plot the two points and draw a line through them

4. Word problem: John is 3 years older than twice the age of his sister Mary. If John is 15 years old, how old is Mary?

   • Solution:
     • Let $x$ represent Mary's age
     • John's age is $2x + 3$
     • Equation: $2x + 3 = 15$
     • Solve for $x$:
       • $2x = 12$
       • $x = 6$
     • Therefore, Mary is 6 years old