6-1 Graphing Systems of Equations

Property

A system of linear equations consists of two or more linear equations that share the same variables and are considered simultaneously.

The solution to a system of linear equations in two variables ($x$ and $y$) is any ordered pair $(x, y)$ that makes BOTH equations true at the same time. Geometrically, this solution is the exact point where the graphs of the two lines intersect.

Examples

• Example 1 (Verifying a Solution): Is the ordered pair $(2, 1)$ a solution to the system $x + y = 3$ and $2x - y = 3$?

Substitute $x=2$ and $y=1$ into both equations:
Equation 1: $2 + 1 = 3$ (True)
Equation 2: $2(2) - 1 = 3 \rightarrow 4 - 1 = 3$ (True)
Because both statements are true, $(2, 1)$ is the solution to the system.

• Example 2 (Not a Solution): Is $(-1, 3)$ a solution to the system $y - x = 4$ and $2x + y = 0$?

Equation 1: $3 - (-1) = 4 \rightarrow 4 = 4$ (True)
Equation 2: $2(-1) + 3 = 0 \rightarrow -2 + 3 = 0 \rightarrow 1 = 0$ (False)
Because it fails the second equation, $(-1, 3)$ is NOT a solution to the system.

Explanation

Think of a single linear equation as a road, and its solutions are all the locations on that road. A system of equations represents two roads on the same map. The "solution to the system" is the crossroads—the single intersection point $(x, y)$ where the two roads meet. This specific pair of numbers is the only location that exists on both roads at the exact same time.

Property

Systems of linear equations are classified by the number of solutions they have, which can be predicted by comparing their slopes ($m$) and y-intercepts ($b$) in slope-intercept form ($y = mx + b$):

• Consistent and Independent (1 Solution): The lines have different slopes. They intersect at exactly one point.
• Inconsistent (No Solution): The lines have the same slope but different y-intercepts. They are parallel lines that never intersect.
• Consistent and Dependent (Infinite Solutions): The lines have the same slope and the same y-intercept. They are coincident (the exact same line), meaning they intersect everywhere.

Examples

• One Solution (Independent): The system $y = 4x + 1$ and $y = 2x + 3$ has different slopes ($4$ and $2$). The lines will cross exactly once, at the point $(1, 5)$.
• No Solution (Inconsistent): The system $y = 3x + 4$ and $y = 3x - 2$ has the same slope ($3$) but different y-intercepts ($4$ and $-2$). The lines are parallel and will never touch.
• Infinite Solutions (Dependent): The system $x + y = 5$ and $3x + 3y = 15$. If you simplify the second equation by dividing everything by 3, you get $x + y = 5$. They represent the exact same line, so every point on the line is a solution.

Explanation