Let . How many solutions does the following system of equations have?
When a system asks for the number of solutions, first simplify any given parameter values. Then compare the equations: nonproportional variable coefficients give one solution, matching ratios throughout give infinitely many solutions, and proportional variable coefficients with different constant ratios give no solution.
Hints
Substitute first
Replace with in both equations before comparing the system.
Look for a multiple
Check whether one equation's variable terms are a constant multiple of the other's variable terms.
Check the constants
If the variable coefficients have a common ratio, compare the constants using that same ratio to classify the system.
Desmos Guide
Use algebra first
Substitution and comparison are faster than graphing for this system. Desmos can verify the result visually.
Enter the simplified lines
Enter the following equations in Desmos:
Compare the graphs
The lines have the same slope but different -intercepts, so Desmos will show two distinct parallel lines.
Step-by-step Explanation
Substitute the value of
Replace with :
Compare the equations
Multiplying the second equation by gives
Its left side now matches the left side of the first equation, but its constant, , does not equal .
Classify the system
The equations have the same slope but different intercepts, so their graphs are distinct parallel lines. Distinct parallel lines do not intersect.