GCSE Maths Simultaneous Equations

Solving Simultaneous Equations

• Simultaneous equations involve finding values of variables that satisfy multiple equations at once.
• The example uses two equations: ( x^2 + y^2 = 13 ) and ( x = (y - 5) ).

Substituting Variables

• Begin by substituting the second equation into the first to eliminate one variable.
• This means replacing ( x ) in the first equation with ( (y - 5) ).

Expanding and Simplifying

• After substitution, the equation becomes ( (y - 5)^2 + y^2 = 13 ).
• Expand and simplify to form a quadratic equation:
  • ( y^2 - 10y + 25 + y^2 = 13 ) leads to ( 2y^2 - 10y + 12 = 0 ).
  • Further simplifying results in ( y^2 - 5y + 6 = 0 ).

Factoring the Quadratic

• The quadratic ( y^2 - 5y + 6 ) can be factored as ( (y - 2)(y - 3) = 0 ).
• Solutions for ( y ) from this equation are ( y = 2 ) and ( y = 3 ).

Finding Corresponding ( x ) Values

• Substitute the ( y ) values back into ( x = (y - 5) ) to find corresponding ( x ) values:
  • For ( y = 2 ): ( x = (2 - 5) = -3 ).
  • For ( y = 3 ): ( x = (3 - 5) = -2 ).

Final Solutions

• The simultaneous equations yield two solutions:
  • ( (x, y) = (-3, 2) )
  • ( (x, y) = (-2, 3) )

Conclusion

• Solving such equations involves substitution, expansion, simplification, and factoring.
• Understanding these steps is crucial for tackling similar problems in mathematics.