Linear Equations in Algebra

Linear Equations

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• The general form of a linear equation is: ax + by = c, where a, b, and c are constants.
• Linear equations can be represented on a graph as a straight line.
• The slope-intercept form of a linear equation is: y = mx + b, where m is the slope and b is the y-intercept.
• The slope represents the rate of change between the x and y variables.
• The y-intercept represents the point at which the line crosses the y-axis.

Surface Area

• Surface area is the total area of the surface of a 3D shape.
• The formula for surface area varies depending on the shape:
  • Rectangular prism: SA = 2lw + 2lh + 2wh
  • Cube: SA = 6s^2
  • Cylinder: SA = 2πr(h + r)
  • Cone: SA = πr(l + r)
• Surface area is measured in square units (e.g. cm^2, m^2).

Volume of 3D Shapes

• Volume is the amount of space inside a 3D shape.
• The formula for volume varies depending on the shape:
  • Rectangular prism: V = lwh
  • Cube: V = s^3
  • Cylinder: V = πr^2h
  • Cone: V = (1/3)πr^2h
• Volume is measured in cubic units (e.g. cm^3, m^3).

Graphing

• Graphing involves plotting points on a coordinate plane to represent relationships between variables.
• The x-axis represents the independent variable, while the y-axis represents the dependent variable.
• Graphs can be used to:
  • Visualize relationships between variables
  • Identify patterns and trends
  • Make predictions and estimates
• Types of graphs:
  • Linear graphs: straight lines
  • Non-linear graphs: curves and shapes

Rational Numbers

• A rational number is a number that can be expressed as the ratio of two integers (e.g. 3/4, 22/7).
• Rational numbers can be represented on a number line.
• Rational numbers can be added, subtracted, multiplied, and divided just like integers.
• Properties of rational numbers:
  • Closure: the result of an operation is always a rational number
  • Commutativity: the order of numbers does not change the result
  • Associativity: the order of operations does not change the result
  • Distributivity: multiplication can be distributed over addition

Linear Equations

• Linear equations have the highest power of the variable(s) as 1.
• The general form of a linear equation is ax + by = c, where a, b, and c are constants.
• Linear equations can be represented on a graph as a straight line.
• The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• The slope represents the rate of change between the x and y variables.
• The y-intercept represents the point at which the line crosses the y-axis.

Surface Area

• Surface area is the total area of the surface of a 3D shape.
• The formula for surface area varies depending on the shape, including:
  • Rectangular prism: SA = 2lw + 2lh + 2wh
  • Cube: SA = 6s^2
  • Cylinder: SA = 2πr(h + r)
  • Cone: SA = πr(l + r)
• Surface area is measured in square units (e.g. cm^2, m^2).

Volume of 3D Shapes

• Volume is the amount of space inside a 3D shape.
• The formula for volume varies depending on the shape, including:
  • Rectangular prism: V = lwh
  • Cube: V = s^3
  • Cylinder: V = πr^2h
  • Cone: V = (1/3)πr^2h
• Volume is measured in cubic units (e.g. cm^3, m^3).

Graphing

• Graphing involves plotting points on a coordinate plane to represent relationships between variables.
• The x-axis represents the independent variable, while the y-axis represents the dependent variable.
• Graphs can be used to:
  • Visualize relationships between variables
  • Identify patterns and trends
  • Make predictions and estimates
• Types of graphs include:
  • Linear graphs: straight lines
  • Non-linear graphs: curves and shapes

Rational Numbers

• A rational number is a number that can be expressed as the ratio of two integers (e.g. 3/4, 22/7).
• Rational numbers can be represented on a number line.
• Rational numbers can be added, subtracted, multiplied, and divided just like integers.
• Properties of rational numbers include:
  • Closure: the result of an operation is always a rational number
  • Commutativity: the order of numbers does not change the result
  • Associativity: the order of operations does not change the result
  • Distributivity: multiplication can be distributed over addition