🔟Elementary Algebra Unit 2 – Linear Equations and Inequalities

Key Concepts and Definitions

• Linear equations represent a straight line on a graph and have the general form $y = mx + b$
• The variable $m$ represents the slope of the line which measures its steepness
  • Slope is calculated as $\frac{\text{rise}}{\text{run}}$ or $\frac{\text{change in y}}{\text{change in x}}$
• The variable $b$ represents the y-intercept, the point where the line crosses the y-axis
• Linear inequalities use the same form as linear equations but include inequality symbols ($<, >, \leq, \geq$)
• A system of linear equations consists of two or more linear equations with the same variables
• The solution to a system of linear equations is the point(s) where the lines intersect
• Parallel lines have the same slope but different y-intercepts and never intersect
• Perpendicular lines have slopes that are negative reciprocals of each other ($m_1 = -\frac{1}{m_2}$)

Linear Equations: Basics and Solving Methods

• To solve a linear equation, isolate the variable on one side of the equation
• Use inverse operations to undo addition, subtraction, multiplication, and division
  • Add or subtract the same value from both sides of the equation
  • Multiply or divide both sides of the equation by the same non-zero value
• Simplify the equation by combining like terms and performing arithmetic operations
• Check your solution by substituting it back into the original equation
• Equations with no solution (inconsistent) result in a false statement ($2 = 5$)
• Equations with infinite solutions (identity) result in a true statement ($3 = 3$)
• Solving equations with fractions may require finding a common denominator
• Literal equations involve solving for a variable other than $x$ or $y$ ($A = \frac{1}{2}bh$, solve for $h$)

Graphing Linear Equations

• To graph a linear equation, find at least two points that satisfy the equation and plot them on a coordinate plane
• Connect the points with a straight line using a ruler
• The x-intercept is the point where the line crosses the x-axis ($y = 0$)
• The y-intercept is the point where the line crosses the y-axis ($x = 0$)
• Horizontal lines have a slope of zero and an equation in the form $y = b$
• Vertical lines have an undefined slope and an equation in the form $x = a$
• The slope-intercept form of a linear equation is $y = mx + b$
  • $m$ represents the slope, and $b$ represents the y-intercept
• The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line

Systems of Linear Equations

• A system of linear equations can be solved using the substitution, elimination, or graphing method
• The substitution method involves solving one equation for a variable and substituting it into the other equation
• The elimination method involves multiplying one or both equations by a constant to eliminate a variable when the equations are added
• The graphing method involves graphing both equations on the same coordinate plane and finding the point of intersection
• A system can have one solution (intersecting lines), no solution (parallel lines), or infinite solutions (coincident lines)
• In a system with no solution, the equations are inconsistent, and the lines do not intersect
• In a system with infinite solutions, the equations are dependent, and the lines coincide (overlap)

Linear Inequalities: Concepts and Solving

• Linear inequalities are solved using the same methods as linear equations, with a few additional rules
• When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed
• The solution to a linear inequality is a range of values rather than a single point
• Inequalities can be combined using the words "and" (intersection) or "or" (union)
  • "And" results in a more restricted solution set
  • "Or" results in a broader solution set
• Compound inequalities involve two inequalities in one statement ($3 < x \leq 7$)
• Absolute value inequalities involve an absolute value expression ($|x - 5| < 3$)
  • Solve by considering two cases: the expression inside the absolute value is positive or negative

Graphing Linear Inequalities

• The solution to a linear inequality is represented by a shaded region on a coordinate plane
• For $y < mx + b$ or $y > mx + b$, the line is dashed (not included in the solution)
• For $y \leq mx + b$ or $y \geq mx + b$, the line is solid (included in the solution)
• Shade above the line for $y > mx + b$ or $y \geq mx + b$
• Shade below the line for $y < mx + b$ or $y \leq mx + b$
• For inequalities with vertical lines ($x < a$ or $x > a$), shade to the left or right of the line
• Systems of linear inequalities involve graphing two or more inequalities on the same coordinate plane
  • The solution is the region where the shaded areas overlap (intersection)

Real-World Applications

• Linear equations can model real-world situations such as cost, revenue, and profit
  • Cost equation: $C = mx + b$, where $m$ is the variable cost and $b$ is the fixed cost
  • Revenue equation: $R = px$, where $p$ is the price per unit and $x$ is the number of units sold
  • Profit equation: $P = R - C$
• Inequalities can represent constraints or limitations in real-world problems
  • Example: A factory must produce at least 100 units but no more than 500 units ($100 \leq x \leq 500$)
• Systems of equations can model situations with multiple unknown quantities
  • Example: A store sells two types of shirts. The total number of shirts sold is 50, and the total revenue is $600. If Type A shirts cost$10 and Type B shirts cost $15, how many of each type were sold?
• Breakeven analysis involves finding the point at which total revenue equals total cost
  • Set the cost equation equal to the revenue equation and solve for the quantity

Common Mistakes and How to Avoid Them

• Forgetting to distribute the negative sign when multiplying or dividing both sides of an equation or inequality by a negative number
  • Double-check signs and distribute carefully
• Incorrectly combining unlike terms ($2x + 3y \neq 5xy$)
  • Only combine terms with the same variables and exponents
• Misinterpreting the slope as the y-intercept or vice versa in the equation $y = mx + b$
  • Remember that $m$ is the slope and $b$ is the y-intercept
• Graphing inequalities incorrectly by shading the wrong side of the line
  • Shade above the line for $y > mx + b$ or $y \geq mx + b$ and below the line for $y < mx + b$ or $y \leq mx + b$
• Failing to reverse the inequality sign when multiplying or dividing by a negative number
  • Always reverse the inequality sign when multiplying or dividing by a negative number
• Confusing the substitution and elimination methods when solving systems of equations
  • Substitution involves solving for a variable, while elimination involves adding equations to cancel out a variable
• Misinterpreting the meaning of "and" and "or" in compound inequalities
  • "And" results in a more restricted solution set (intersection), while "or" results in a broader solution set (union)