The Main Idea

• Equation Basics:
  • Contains an equal sign (=) separating two expressions
  • May include coefficients, variables, terms, and expressions
  • Goal: Isolate the variable on one side
• Types of Equations:
  • Simple equations (solved mentally): [latex]2x = 6[/latex]
  • Multi-step equations (require multiple operations): [latex]4(\frac{1}{3}t + \frac{1}{2}) = 6[/latex]
  • Equations with variables on both sides: [latex]4x - 6 = 2x + 10[/latex]
• Key Properties:
  • Addition Property of Equality: If [latex]a = b[/latex], then [latex]a + c = b + c[/latex]
  • Multiplication Property of Equality: If [latex]a = b[/latex], then [latex]ac = bc[/latex]
  • These properties maintain the balance of the equation
• Solving Strategy:
  • Simplify expressions on both sides
  • Isolate variable terms on one side
  • Isolate the variable itself
• Common Techniques:
  • Combining like terms
  • Clearing fractions or decimals
  • Moving terms between sides of the equation

Solving Process

1. Simplify each side of the equation:
   • Clear parentheses
   • Combine like terms
2. Move variable terms to one side:
   • Add or subtract to move terms
   • Choose the side with the larger coefficient (if applicable)
3. Move constant terms to the other side:
   • Add or subtract as needed
4. Isolate the variable:
   • Multiply or divide both sides
5. Check the solution:
   • Substitute the result back into the original equation

The Main Idea

• Definition:
  • Equation form: [latex]y = mx + b[/latex]
  • [latex]m[/latex] represents the slope
  • [latex]b[/latex] represents the y-intercept
• Slope ([latex]m[/latex]):
  • Measures the steepness of the line
  • Calculated as [latex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/latex]
  • Positive slope: line rises from left to right
  • Negative slope: line falls from left to right
  • Zero slope: horizontal line
  • Undefined slope: vertical line
• Y-intercept ([latex]b[/latex]):
  • Point where the line crosses the y-axis
  • Occurs when [latex]x = 0[/latex]
  • Represents the starting point of the line on the y-axis
• Interpreting the equation:
  • [latex]m[/latex] tells how much [latex]y[/latex] changes for each unit increase in [latex]x[/latex]
  • [latex]b[/latex] tells the value of [latex]y[/latex] when [latex]x = 0[/latex]

The Main Idea

• Definition:
  • Equation form: [latex]y - y_1 = m(x - x_1)[/latex]
  • [latex]m[/latex] is the slope of the line
  • [latex](x_1, y_1)[/latex] is a known point on the line
• Components:
  • Slope ([latex]m[/latex]): Rate of change of the line
  • Point [latex](x_1, y_1)[/latex]: Any known point on the line
• Usage:
  • Useful when given a point and slope
  • Can be easily converted to slope-intercept form
  • Often used in calculus for tangent line equations
• Flexibility:
  • Any point on the line can be used as [latex](x_1, y_1)[/latex]
  • Allows for easy verification of points on the line

The Main Idea

• Definition:
  • Equation form: [latex]Ax + By = C[/latex]
  • [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex] are integers
  • [latex]A[/latex] and [latex]B[/latex] are not both zero
• Components:
  • [latex]A[/latex]: Coefficient of x
  • [latex]B[/latex]: Coefficient of y
  • [latex]C[/latex]: Constant term
• Characteristics:
  • All terms are on one side of the equation
  • x and y terms are on the left, constant on the right
  • Coefficients are typically reduced to lowest terms
• Conversions:
  • Can be converted from/to slope-intercept form
  • Can be converted from/to point-slope form

The Main Idea

• Vertical Lines:
  • Equation: [latex]x = c[/latex], where [latex]c[/latex] is a constant
  • Slope: Undefined
  • All points have the same x-coordinate
  • Cannot be expressed in slope-intercept form
  • Represent a constant [latex]x[/latex]-value for all [latex]y[/latex]-values
• Horizontal Lines:
  • Equation: [latex]y = c[/latex], where [latex]c[/latex] is a constant
  • Slope: Zero
  • All points have the same [latex]y[/latex]-coordinate
  • Can be expressed in slope-intercept form with [latex]m = 0[/latex]
  • Represent a constant [latex]y[/latex]-value for all [latex]x[/latex]-values
• Characteristics:
  • Vertical lines are parallel to the y-axis
  • Horizontal lines are parallel to the x-axis
  • These lines are perpendicular to each other
• Identifying Line Type:
  • If all points share the same [latex]x[/latex]-coordinate: Vertical line
  • If all points share the same [latex]y[/latex]-coordinate: Horizontal line
  • If neither: Oblique (slanted) line

The Main Idea

• Parallel Lines:
  • Equation: Lines have the same slope but different y-intercepts.
  • Slope: Same for all parallel lines.
  • Properties:
    • Parallel lines never intersect.
    • The distance between the lines is constant (equidistant).
• Perpendicular Lines:
  • Equation: Lines have slopes that are negative reciprocals of each other.
  • Slope: The product of the slopes of two perpendicular lines is -1.
  • Properties:
    • Perpendicular lines intersect at a 90-degree angle.
    • If one line has slope [latex]m[/latex], the perpendicular line will have slope [latex]-\dfrac{1}{m}[/latex].
• Identifying Line Type:
  • Parallel Lines: Look for lines with identical slopes and different y-intercepts.
  • Perpendicular Lines: Check if the product of their slopes is -1.
• Writing equations of parallel and perpendicular lines
  • To write the equation of a line parallel to a given line, use the same slope as the original line but adjust the y-intercept based on a point the line passes through.
  • To write the equation of a line perpendicular to a given line, use the negative reciprocal of the original slope and find the y-intercept using a point the line passes through.