📖Definition
A system of equations is a set of two or more equations with the same variables. The solution is the point(s) where all equations are true simultaneously. For linear systems, this is where the lines intersect on a graph.
📐Formula
A system of two linear equations can have one solution (lines intersect), no solution (parallel lines), or infinitely many solutions (same line).
📝Step-by-Step Guide
Substitution Method
Solve one equation for one variable, then substitute into the other equation.
Elimination Method
Add or subtract equations to eliminate one variable.
Solve for First Variable
Use algebra to find the value of one variable.
Find the Second Variable
Substitute back into either original equation to find the other variable. Check in both equations.
⚠️Common Mistakes to Avoid
- Arithmetic errors when eliminating variables
- Substituting into the wrong equation
- Forgetting to find both variables
- Not checking the solution in both equations
- Misidentifying parallel lines (no solution)
✏️Practice Problems
Solve: x + y = 10, x - y = 4
Answer: x = 7, y = 3
Solve: 2x + 3y = 12, x - y = 1
Answer: x = 3, y = 2
Solve: 3x - 2y = 8, 2x + 5y = -1
Answer: x = 2, y = -1
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