GCSE Maths Simultaneous Equations

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.

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!