Example: Solve the system

$x^2 + y^2 = 9 \quad \text{(Equation A)}$

$x = y - 5 \quad \text{(Equation B)}$

1. Rearrange Equation B to isolate $x$:

• Equation B: $x = y - 5$

2. Substitute $x$ from Equation B into Equation A:

• Substitute $x = y - 5$ into $x^2 + y^2 = 9$:
• $(y - 5)^2 + y^2 = 9$
• $y^2 - 10y + 25 + y^2 = 9$
• $2y^2 - 10y + 25 = 9$
• $2y^2 - 10y + 16 = 0$
• $y^2 - 5y + 8 = 0$

1. Solve the quadratic equation for $y$:

• Factorize the quadratic:
• $(y - 7)(y - 4) = 0$
• $y = 7 \quad \text{or} \quad y = 4$

3. Substitute back to find $x$:

• If $y = 7$:
• $x = 7 - 5 = 2$
• If $y = 4$:
• $x = 4 - 5 = -1$

2. Solutions:

• $(2, 7)$
• $(-1, 4)$

Calculator Method

1. Use the quadratic equation solver to find the roots of the quadratic:

• $y = 7$
• $y = 4$ In a question that does not require full working, this is a valid method.

Q3 (Jan 2013, Q4)

(i) Solve the simultaneous equations:

$y = 2x^2 - 3x - 5$

$10x + 2y + 11 = 0$

Q3 (Jan 2013, Q4)

(ii) What can you deduce from the answer to part (i) about the curve $y$ $=$ $2x^2 - 3x - 5$ and the line $10x + 2y + 11 = 0$?