This section of our PERT Study Guide covers systems. When given more than one equation, we can use the equations together to solve for variables that would be impossible to solve for with a single equation. We can also solve a system of equations graphically, or see where the two equations intersect when plotted on a coordinate plane.
Using the Substitution Method
In the substitution method, we solve one of the equations in the system for one variable and then substitute this expression into the other equation. This reduces the system to a single equation with one variable, which is then solvable using basic algebra.
Solving a System Using Substitution
- Isolate a variable: Choose one of the equations in the system and solve it for one of the variables.
- Substitute: Substitute the isolated expression into the other equation. This will leave you with an equation that has only one variable.
- Solve: Solve the resulting equation for the remaining variable.
- Back substitute: Substitute the value you found back into the first equation to solve for the other variable.
Example
Solve the system $y = 2x + 3 \, $ and $ \, 3x − y = 7$.
Substitute $y = 2x + 3$ into the second equation:
$3x − (2x + 3) = 7$
Then simplify and solve:
$x = 10$
Back substitute $x = 10$ to find $y$, resulting in $y = 23$. The solution is $(10, 23)$, meaning $x$ is equivalent to $10$ and $y$ is equivalent to $23$.
Using the Elimination Method
The elimination method involves adding or subtracting the equations in a system to eliminate one of the variables, making it possible to solve for the remaining variable.
Solving a System Using Elimination
- Align equations: Write both equations in standard form: $Ax + By = C$
- Equalize coefficients: If necessary, multiply one or both equations by a constant to make the coefficients of one variable the same (or opposites).
- Eliminate: Add or subtract the equations to eliminate one variable.
- Solve: Solve the resulting equation for the remaining variable.
- Back substitute: Substitute the solution back into one of the original equations to find the other variable.
Example
Solve the system $2x + 3y = 8$ and $4x − 3y = 16$.
Adding the equations eliminates $y$, as $3y$ and $−3y$ cancel out. This leaves us with $6x = 24$.
Solving for $x$ gives us $x = 4$. We can insert $4$ in for $x$ in either of the equations to find $y$.
$2(4) + 3y = 8$
$8 + 3y = 8$
$3y = 0$
$y = 0$
Solution: $(4, 0)$
Using a Graphical Solution
The graphical solution method involves graphing both equations on a coordinate plane to visually identify the point where they intersect.
Understanding Possible Outcomes
- One solution: If the lines intersect at one point, the system has a unique solution at that point (consistent and independent).
- No solution: If the lines are parallel, the system has no solution (inconsistent).
- Infinite solutions: If the lines overlap completely, they have all points in common (consistent and dependent).
Example
For the equations $y = 2x + 1$ and $\, y = −x +4$, graphing reveals an intersection at $(1, 3)$, which is the solution.
Key Tip: Use substitution when one equation is already solved for a variable. Use elimination when adding or subtracting will quickly remove a variable. Graphing is best for visual understanding but may be less precise for exact solutions.
Key Tip: Always double-check your solutions. After solving, substitute your solutions back into the original equations to verify accuracy.