Understanding Linear Equations

Standard Form

Any linear equation can be written as ( y = mx + b ).
- m: The slope of the line.
- b: The y-intercept.
- Slope ( m ) is the rise over run, or the inclination of the line.

Set ( x = 0 ) in the equation to find the y-intercept.
- Equation becomes ( y = m \times 0 + b ), simplifies to ( y = b ).
- Therefore, the line intercepts the y-axis at (0, b).

Use two points to verify:
- ( y = m \times 1 + b ), results in the point (1, m + b).
- Slope ( m = \frac{(m+b) - b}{1 - 0} = m ).

Line A
- Arbitrary starting point, calculate slope:
- ( \Delta x = 3, \Delta y = -2 ) results in slope ( m = \frac{-2}{3} ).
- Y-intercept calculated as ( b = \frac{4}{3} ).
- Equation: ( y = -\frac{2}{3}x + \frac{4}{3} ).
Line B
- ( \Delta x = 1, \Delta y = 3 ) results in slope ( m = 3 ).
- Y-intercept ( b = 1 ).
- Equation: ( y = 3x + 1 ).
Line C
- Y-intercept ( b = -2 ).
- ( \Delta x = 4, \Delta y = 2 ) results in slope ( m = \frac{1}{2} ).
- Equation: ( y = \frac{1}{2}x - 2 ).

Equation: ( y = 2x + 5 )
- Y-intercept: 5
- Slope: 2 (For every 1 unit right, move up 2 units)
- Graph is a line moving through these coordinates.
Equation: ( y = -0.2x + 7 ) or ( y = -\frac{1}{5}x + 7 )
- Y-intercept: 7
- Slope: ( -\frac{1}{5} ) (For every 5 units right, move down 1 unit)
- Line is downward sloping.
Equation: ( y = -x )
- Y-intercept: 0
- Slope: -1 (For every 1 unit right, move down 1 unit)
- Line passes through the origin.
Equation: ( y = 3.75 )
- No x term, slope is 0.
- Y-intercept: 3.75
- Horizontal line at ( y = 3.75 ).

Linear equations can be graphed by identifying the slope (m) and y-intercept (b).
Converting equations from standard form to visual graphs helps understand the relationship between x and y.
Each linear equation describes a straight line on a graph, determined by its slope and y-intercept.