Lesson 4: Solving Special Systems of Linear Equations

Property

To solve special systems of linear equations by graphing, follow these steps:

1. Graph the first equation.
2. Graph the second equation on the same rectangular coordinate system.
3. Determine whether the lines intersect at one point, are parallel (no intersection), or are the same line (infinite intersections).
4. Classify the system: If the lines intersect at one point, the system has exactly one solution. If the lines are parallel, there is no solution. If the lines are identical, there are infinitely many solutions.

Examples

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

Property

If solving a system of equations by substitution results in a true statement without variables, such as $0=0$, the equations are dependent. The graphs of the two equations are the same line, and the system has infinitely many solutions.

Examples

• Solve the system $\begin{cases} y = 2x - 3 \\ 4x - 2y = 6 \end{cases}$. Substitute $y$ in the second equation: $4x - 2(2x-3) = 6$. This simplifies to $4x - 4x + 6 = 6$, which gives $6=6$. This is a true statement, so there are infinitely many solutions.

• Solve the system $\begin{cases} x = 3y + 1 \\ 2x - 6y = 2 \end{cases}$. Substitute $x$ in the second equation: $2(3y+1) - 6y = 2$. This simplifies to $6y + 2 - 6y = 2$, which results in $2=2$. This is a true statement, indicating infinitely many solutions.