Lines and Quadratics
3.2 Systems of Equations
What a Solution Means
A solution to a system is a point that satisfies every equation at the same time.
Definition: System of Linear Equations
A system is a set of equations solved simultaneously.
Fact: Number of solutions
For two linear equations in two variables:
- One solution: lines intersect once.
- No solution: lines are parallel.
- Infinitely many solutions: same line.
Example: Solve by substitution
Solve:
$$\begin{cases} y=2x+1 \ 3x+y=11 \end{cases}$$
Show Solution
Substitute $y=2x+1$ into second equation:
$$3x+(2x+1)=11\Rightarrow 5x+1=11\Rightarrow 5x=10\Rightarrow x=2.$$
Then
$$y=2(2)+1=5.$$
Solution: $(2,5)$.
MyOpenMath: Practice: substitution method (placeholder)
Elimination Method
Elimination is usually faster when coefficients line up nicely.
Example: Solve by elimination
Solve:
$$\begin{cases} 2x+3y=7 \ 4x-3y=5 \end{cases}$$
Show Solution
Add equations to eliminate $y$:
$$6x=12\Rightarrow x=2.$$
Substitute into first equation:
$$2(2)+3y=7\Rightarrow 4+3y=7\Rightarrow y=1.$$
Solution: $(2,1)$.
