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GCSE Maths Simultaneous Equations
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GCSE Maths Simultaneous Equations

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Questions and Answers

How would you substitute the value of x in terms of y into the equation x^2 + y^2 = 13?

(y - 5)^2 + y^2 = 13

What is the simplified quadratic equation obtained after expansion?

y^2 - 5y + 6 = 0

List the two possible values of y obtained from factoring the quadratic equation.

y = 2, y = 3

What are the corresponding x values for both y values obtained?

<p>For y = 2, x = -3; for y = 3, x = -2</p> Signup and view all the answers

Summarize the final solutions for the simultaneous equations.

<p>(x, y) = (-3, 2) and (-2, 3)</p> Signup and view all the answers

Study Notes

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.

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Description

Test your skills on simultaneous equations with this GCSE Maths quiz. You will solve a system of equations where one variable is expressed in terms of another. Get ready to factor quadratics and find the solutions!

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