How to Solve Special Systems
A special system of \(2\) equations is a system of equations that can be solved using a specific method, such as substitution or elimination. Two common types of special systems are Consistent, dependent systems and Consistent, independent systems.
Tutor-style math help
Solve Special Systems: what to notice and how to work it
Systems skill
A system asks for values that make every equation true at the same time. On a graph, the solution is where the graphs meet.
What to notice first
Choose the method based on the form you are given. Substitution is friendly when a variable is isolated; elimination is friendly when coefficients line up.
Common student mistake
Do not stop after finding one variable. A two-variable system usually needs an ordered pair, and that pair must check in every original equation.
Key formulas and cues
\(\text{linear system solution}=(x,y)\)
\(\text{same slope, different intercepts}\Rightarrow\text{no solution}\)
\(\text{same line}\Rightarrow\text{infinitely many solutions}\)
A reliable path
- Choose a methodGraph, substitute, or eliminate depending on the form.
- Solve one variableUse the cleanest equation to find one value.
- Find and check the pairSubstitute back and verify both equations.
Worked examples
Substitution
Example: \(y=x+2\) and \(y=2x-1\)
- Set the right sides equal.
- Solve x + 2 = 2x – 1 to get x = 3.
- Substitute to find y.
Answer: \((3,5)\)
Elimination
Example: \(x+y=8\), \(x-y=2\)
- Add the equations to eliminate y.
- 2x = 10, so x = 5.
- Use x + y = 8 to find y = 3.
Answer: \((5,3)\)
Try one before moving on
Try: Solve \(y=x+1\) and \(y=3x-3\).
Answer: \((2,3)\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Solve Special Systems: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. The solution to two linear equations is usually:
2. Two parallel lines have:
3. A system answer should be checked in:
Related Topics
- How to Solve Systems of Equations Graphically
- How to Solve Systems of Linear Inequalities
- How to Solve Systems of Equations with Substitution
- How to Find the Number of Solutions in a System of Equations
Step-by-step to solve special systems
To solve special systems, follow the step-by-step guide below:
- Consistent, dependent systems: A consistent, dependent system of \(2\) equations is a system of equations that has an infinite number of solutions. This occurs when the two equations are identical or one equation is a multiple of the other. For example, the system of equations \(2x + 3y = 6\) and \(4x + 6y = 12\) has an infinite number of solutions because the second equation is simply twice the first equation. In this case, we can solve the system by solving either equation for one variable and then substituting it into the other equation.
- Consistent, independent systems: A consistent, independent system of \(2\) equations is a system of equations that has one unique solution. This occurs when the two equations are not identical and the system of equations can be solved by elimination or substitution.
Solving Special Systems, Example:
Solve the system of equations
\(\begin{cases}2x+3y=6\\ 4x+6y=12\end{cases}\)
Solution:
Using elimination method: Add the two equations \(\begin{array}{r} &2x + 3y = 6\\ +\!\!\!\!\!\!&4x + 6y = 12 \\ \hline \end{array} \\ \ \ \ \ \ \ 6x + 9y = 18\)
So, we can see that \(x = -3\) then we can substitute this value in one of the equations
\(2(-3) + 3y = 6\)→\(y = 3\)
So, the solution of this system is \(x = -3\) and \(y = 3\)
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