## Podcast Beta

## 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?

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Summarize the final solutions for the simultaneous equations.

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## 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!