How do you solve the simultaneous equations x²+y²=5 and y=3x+1 ?

Apr 1, 2018

$x = \frac{2}{5} , \quad y = \frac{11}{5}$

or

$x = - 1 , y = 2$

Explanation:

Substitute $y = 3 x + 1$ in ${x}^{2} + {y}^{2} = 5$ to obtain

${x}^{2} + {\left(3 x + 1\right)}^{2} = 5 q \quad \implies q \quad 10 {x}^{2} + 6 x - 4 = 0$

Hence

$5 {x}^{2} + 3 x - 2 = 0 q \quad \implies q \quad 5 {x}^{2} + 5 x - 2 x - 2 = 0 q \quad \implies$
$5 x \left(x + 1\right) - 2 \left(x + 1\right) = 0 q \quad \implies q \quad \left(5 x - 2\right) \left(x + 1\right) = 0$

Hence

• either $x = \frac{2}{5}$, $y = 3 \times \frac{2}{5} + 1 = \frac{11}{5}$
• or $x = - 1$, $y = 3 \times \left(- 1\right) + 1 = - 2$