Linear Equations in Algebra

Study Notes

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 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 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 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

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 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 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 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 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

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