Integrated Math 1 / Unit 6: Systems of Equations and Inequalities
6.4 Special Cases and Choosing a Method
Notes
Pick the method that makes the work easiest.
- Graphing: good for estimates. Substitution: good when a variable is isolated. Elimination: good when equations are in standard form.
- If solving gives a false statement (like $0 = 5$), there is no solution. A true statement (like $0 = 0$) means infinitely many solutions.
- Equations with equal slopes and equal intercepts are the same line.
- Example: $x = 5y$ and $2x - 3y = 14$: $10y - 3y = 14$, so $y = 2$ and $x = 10$.
Practice
How many solutions does the system $2x + y = 5$ and $4x + 2y = 10$ have?
The second equation is twice the first, so both describe the same line.
How many solutions does $x + y = 3$ and $x + y = 8$ have?
The same expression cannot equal both 3 and 8.
Solve: $x = 5y$ and $2x - 3y = 14$.
$2(5y) - 3y = 14 \Rightarrow 7y = 14 \Rightarrow y = 2$, so $x = 10$.
Ready for more? Take the Unit 6 test.