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Solve sets of two or more linear equations simultaneously using methods like substitution and elimination.

Multiple Unknowns: Systems of Equations

A system of equations is a set of two or more equations with the same variables, and solving the system means finding values that satisfy all equations simultaneously. Systems appear whenever two conditions must be met at once β€” comparing phone plans, finding where two paths cross, or balancing chemical reactions.

Components of Systems of Equations

This section covers the main methods for solving systems:

  • Graphing Method: Graph both equations on the same coordinate plane; the intersection point is the solution.
  • Substitution Method: Solve one equation for a variable, then substitute that expression into the other equation.
  • Elimination Method: Add or subtract equations (after multiplying if needed) to eliminate one variable, then solve for the other.
  • Types of Solutions: A system has one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line).

Examples of Systems of Equations

Graphing Method Examples

  • Solve y = x + 1 and y = -x + 5: Graph both lines. They intersect at (2, 3), so x = 2 and y = 3.
  • Solve y = 2x and y = x + 3: The lines cross where 2x = x + 3, giving x = 3 and y = 6. The intersection is (3, 6).
  • Two friends start at different positions and walk toward each other. Their paths, modeled as linear equations, cross at the point where they meet.

Substitution Method Examples

  • Solve y = 3x and 2x + y = 10: Substitute 3x for y: 2x + 3x = 10, so 5x = 10 and x = 2. Then y = 3(2) = 6.
  • Solve x = y - 4 and 3x + 2y = 17: Substitute (y - 4) for x: 3(y - 4) + 2y = 17, so 3y - 12 + 2y = 17, 5y = 29, y = 29/5. Then x = 29/5 - 4 = 9/5.
  • Solve y = 2x + 1 and 4x - y = 5: Substitute: 4x - (2x + 1) = 5, so 2x - 1 = 5, 2x = 6, x = 3. Then y = 2(3) + 1 = 7.

Elimination Method Examples

  • Solve 2x + y = 7 and 3x - y = 8: Add the equations to eliminate y: 5x = 15, so x = 3. Then 2(3) + y = 7, y = 1.
  • Solve x + 2y = 10 and 3x + 2y = 18: Subtract the first from the second: 2x = 8, so x = 4. Then 4 + 2y = 10, 2y = 6, y = 3.
  • Solve 2x + 3y = 12 and 4x - 3y = 6: Add to eliminate y: 6x = 18, x = 3. Then 2(3) + 3y = 12, 3y = 6, y = 2.

Types of Solutions Examples

  • y = 2x + 1 and y = 2x - 3 are parallel (same slope, different intercepts), so there is no solution.
  • y = 3x + 2 and 6x - 2y = -4 are the same line (rearrange the second: y = 3x + 2), so there are infinitely many solutions.
  • y = x + 4 and y = -x + 2 have different slopes, so they intersect at exactly one point: (βˆ’1, 3).