The Equation of a Line in the Plane

An equation of the form:

\[ 4x + 3y = 12 \]

contains two unknowns, \( x \) and \( y \). A solution to this equation is a pair \((x, y)\) that satisfies the equation. For example, \( (3,0) \) is a solution because when \( x = 3 \), substituting it into the equation gives:

\[ 4(3) + 3(0) = 12 \]

But there are many more solutions, such as \( (0,4) \) and \( (-3,8) \).

In general, you can freely choose one of the unknowns, and then solve for the other. When you plot all the solutions in a coordinate system, they form a straight line. This means the equation represents all points \((x, y)\) whose coordinates satisfy the equation.

For example, the line given by \( 4x + 3y = 12 \) passes through the points \( (3,0) \) and \( (0,4) \).

Lines Divide the Plane

The equation of a line also divides the plane into two regions, or half-planes:

• Points in one half-plane satisfy \( 4x + 3y > 12 \).
• Points in the other half-plane satisfy \( 4x + 3y < 12 \).

To determine which inequality applies to which half-plane, you can test a point. The easiest choice is the origin \((0,0)\).

Substituting \( x = 0 \) and \( y = 0 \) into the equation:

\[ 4(0) + 3(0) = 0 \]

Since \( 0 < 12 \), the region containing the origin satisfies \( 4x + 3y < 12 \). The other half-plane, where \( 4x + 3y > 12 \), is on the opposite side of the line.

General Form of a Line

Any equation of the form:

\[ ax + by = c \]

represents a straight line, provided that \( a \) and \( b \) are not both zero. If you multiply both sides of the equation by the same nonzero number, you still get the same line.

For example, the equations:

\[ 4x + 3y = 12 \]

\[ 8x + 6y = 24 \]

represent the same line, since the second equation is just the first one multiplied by 2.

Intercepts of a Line

To find where a line crosses the y-axis, set \( x = 0 \):

\[ 4(0) + 3y = 12 \]

\[ 3y = 12 \]

\[ y = 4 \]

So, the y-intercept is \( (0,4) \).

To find where the line crosses the x-axis, set \( y = 0 \):

\[ 4x + 3(0) = 12 \]

\[ 4x = 12 \]

\[ x = 3 \]

So, the x-intercept is \( (3,0) \).

A line is horizontal when \( a = 0 \), meaning it has an equation like \( y = c \). A line is vertical when \( b = 0 \), meaning it has an equation like \( x = c \).