Linear Systems of Equations with Infinitely Many Solutions
A linear system has infinitely many solutions if the graphs of the two
equations coincide.
That is, if the graphs are identical, the linear system has infinitely many
solutions.
Example 1
Graph each equation to find the solution of this system.
Solution
To graph each equation, first write it in slope-intercept form.
• In slope-intercept form, the first equation is y = -2x - 3.
The y-intercept is (0, -3). Plot (0, -3).
Use the slope,
, to plot a second point.
Draw the line through the two points.
• In slope-intercept form, the second equation is y = -2x - 3.
This equation is identical to the first equation.
Therefore, its graph is identical to that of the first equation.
Thus, the lines coincide.
Because the lines coincide they have infinitely many points in common.
Thus, the system has infinitely many solutions.
These solutions may be stated in several ways, including:
• “The set of all ordered pairs for which 2y + 6 = -4x.â€
• “The set of all ordered pairs for which 2x + y = -3.â€
• “The set of all ordered pairs for which y = -2x - 3.â€
If the graphs of the equations of a system are identical, as in
the previous example, the equations are called dependent. Otherwise, the
equations are called independent.