Linear Equations in Algebra

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Questions and Answers

What is the general form of a linear equation?

  • ax^2 + by = c
  • ax - by = c
  • ax + by = c (correct)
  • ax + by = 0

The surface area of a cube is the same as the volume of a cube.

False (B)

What is the formula for the surface area of a rectangular prism?

2lw + 2lh + 2wh

The slope of a linear equation represents the rate of change between the ______________ and y variables.

<p>x</p> Signup and view all the answers

What is the unit of measurement for surface area?

<p>square units (A)</p> Signup and view all the answers

A linear graph is a curve or shape.

<p>False (B)</p> Signup and view all the answers

What is the formula for the volume of a cylinder?

<p>Ï€r^2h</p> Signup and view all the answers

A ______________ number is a number that can be expressed as the ratio of two integers.

<p>rational</p> Signup and view all the answers

Match the following shapes with their corresponding formulas for volume:

<p>Rectangular prism = V = lwh Cube = V = s^3 Cylinder = V = πr^2h Cone = V = (1/3)πr^2h</p> Signup and view all the answers

Rational numbers can be represented on a coordinate plane.

<p>False (B)</p> Signup and view all the answers

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Study Notes

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 = 2Ï€r(h + r)
    • Cone: SA = Ï€r(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 = Ï€r^2h
    • Cone: V = (1/3)Ï€r^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 = 2Ï€r(h + r)
    • Cone: SA = Ï€r(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 = Ï€r^2h
    • Cone: V = (1/3)Ï€r^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

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