General Equation of a Line: \( ax + by = c \)

The general equation of a line in a two-dimensional \(x\)-\(y\) plane may be written as: \[ a x + b y = c \] The coefficients \(a\), \(b\), and \(c\) determine the properties of the line, giving information on whether the line is horizontal, vertical, or slanted, as well as its \(x\)- and \(y\)-intercepts.

Example 1: Finding Ordered Pair Solutions

a) Find four ordered pairs that are solutions to the linear equation in two variables given by: \( y + 2x = 1 \)
b) Plot the ordered pairs found in part a) and join the points obtained.

View Solution to Example 1

a) Because the given equation \( y + 2x = 1 \) has two variables \(x\) and \(y\), it has an infinite number of ordered pair solutions \((x, y)\) that can be found by assigning a value to one variable and solving for the other:

  1. Let \(x = 2\): substitute into \(y + 2x = 1 \implies y + 2(2) = 1 \implies y + 4 = 1 \implies y = -3\). The ordered pair \((2, -3)\) is a solution.
  2. Let \(x = -1\): substitute into \(y + 2x = 1 \implies y + 2(-1) = 1 \implies y - 2 = 1 \implies y = 3\). The ordered pair \((-1, 3)\) is a solution.
  3. Let \(y = 5\): substitute into \(y + 2x = 1 \implies 5 + 2x = 1 \implies 2x = -4 \implies x = -2\). The ordered pair \((-2, 5)\) is a solution.
  4. Let \(y = -1\): substitute into \(y + 2x = 1 \implies -1 + 2x = 1 \implies 2x = 2 \implies x = 1\). The ordered pair \((1, -1)\) is a solution.

Additional solutions include: \((0, 1), \left(\frac{1}{2}, 0\right), (3, -5), \left(-\frac{1}{2}, 2\right)\). You can verify all pairs via substitution.

b) Plotting these ordered pairs and joining them yields a slanted line.

Plots of ordered pairs
Fig.1 - Graph of a slanted line from ordered pairs

Conclusion: The graph of any equation of the form \( ax + by = c \) is a line.

Horizontal Lines: \( a = 0 \)

If the coefficient \(a\) in the linear equation \(ax + by = c\) is equal to zero, the equation becomes: \[ 0x + by = c \implies by = c \] Dividing all terms by \(b\) yields: \[ y = \frac{c}{b} = B \] The ordered pairs that satisfy this equation are of the form \((k, B)\), where \(k\) is any real number. These form a horizontal line.

View Example 2 (Horizontal Lines)

Example 2:
a) Find four ordered pairs that are solutions to \( y = 3 \).
b) Plot the ordered pairs and join them.
c) Give two more examples of horizontal lines.

Solution:

a) Written as \( 0x + y = 3 \), any ordered pair of the form \((k, 3)\) is a solution. Examples: \((0, 3), (-2, 3), (3, 3), (4, 3)\).

b) Plotting these points forms a horizontal line passing through \(y = 3\).

Plots of ordered pairs in a horizontal line
Fig.2 - Horizontal line graph

c) More examples: \( y = -3 \) (horizontal line passing through \(y = -3\)) and \( y = 0 \) (the \(x\)-axis).

Vertical Lines: \( b = 0 \)

If the coefficient \(b\) in the linear equation \(ax + by = c\) is equal to zero, the equation becomes: \[ ax + 0y = c \implies ax = c \] Dividing all terms by \(a\) yields: \[ x = \frac{c}{a} = A \] The ordered pairs that satisfy this equation are of the form \((A, k)\), where \(k\) is any real number. These form a vertical line.

View Example 3 (Vertical Lines)

Example 3:
a) Find four ordered pairs that are solutions to \( x = -2 \).
b) Plot the ordered pairs and join them.
c) Give two more examples of vertical lines.

Solution:

a) Written as \( x + 0y = -2 \), any ordered pair of the form \((-2, k)\) is a solution. Examples: \((-2, 3), (-2, -3), (-2, 1), (-2, 0)\).

b) Plotting these points forms a vertical line passing through \(x = -2\).

Plots of ordered pairs in a vertical line
Fig.3 - Vertical line graph

c) More examples: \( x = 3 \) (vertical line passing through \(x = 3\)) and \( x = 0 \) (the \(y\)-axis).

\(x\)- and \(y\)-Intercepts of the Graph of a Line

For general linear equations \( ax + by = c \) where neither \(a\) nor \(b\) is zero:

View Example 4 (Finding Intercepts)

Example 4: Find the \(x\)- and \(y\)-intercepts of:
a) \( 2x - y = 2 \)
b) \( 4x + 2y = 0 \)

Solution:

a) For \( 2x - y = 2 \):
- Set \(y = 0 \implies 2x = 2 \implies x = 1\) (\(x\)-intercept: \((1, 0)\)).
- Set \(x = 0 \implies -y = 2 \implies y = -2\) (\(y\)-intercept: \((0, -2)\)).

b) For \( 4x + 2y = 0 \):
- Set \(y = 0 \implies 4x = 0 \implies x = 0\) (\(x\)-intercept: \((0, 0)\)).
- Set \(x = 0 \implies 2y = 0 \implies y = 0\) (\(y\)-intercept: \((0, 0)\)).
Note: Any equation of the form \( ax + by = 0 \) passes through the origin \((0, 0)\).

Graphs, x and y intercepts of the lines: 4x + 2y = 0 , 2x - y = 2
Fig.4 - Intercepts of linear graphs

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