Solving Systems of Equations
A system of equations is a set of two or more equations with the same variables. The solution is the set of variable values that satisfy all equations simultaneously.
There are several methods to solve systems, including substitution, elimination, and graphical methods. Here, we'll use the elimination method.
Worked Example
Solve the system:
\[\begin{cases} 2x + 3y = 13 \\ 4x - 3y = 5 \end{cases}\]
Step 1: Add the equations to eliminate $y$
\[\begin{align*} (2x + 3y) + (4x - 3y) &= 13 + 5 \\ 2x + 4x + 3y - 3y &= 18 \\ 6x &= 18 \\ x &= \frac{18}{6} = 3 \end{align*}\]
Step 2: Substitute $x = 3$ into the first equation
\[\begin{align*} 2x + 3y &= 13 \\ 2(3) + 3y &= 13 \\ 6 + 3y &= 13 \\ 3y &= 13 - 6 = 7 \\ y &= \frac{7}{3} \end{align*}\]
Solution: $x = 3$, $y = \frac{7}{3}$
Takeaways
- Systems of equations can be solved by substitution, elimination, or graphing.
- The elimination method involves adding or subtracting equations to eliminate a variable.
- Always check your solution by substituting back into the original equations.