How to Solve the System of Quadratic Equations
Algebra math problem
Solve the system of quadratic equations x² + y² = 25 and 2x + y = 10, then find the value of x + y
x^2+y^2=25
2x+y=10
x+y=?
Solution to the system of quadratic equations
Let
x² + y² = 25………………eq(1)
2x + y = 10………………eq(2)
From equation 2
2x + y = 10
⇒ y = 10 – 2x……………….eq(3)
From equation 1 and equation 3
x² + y² = 25
⇒ x² + (10 – 2x)² = 25
⇒ x² + 100 – 40x + 4x² = 25
so, 5x² – 40x + 75 = 0
⇒ x² – 8x + 15 = 0
x² – 8x + 15 = 0 is a quadratic equation so we can apply quadratic equation formula
\begin{aligned} x&=\dfrac{8 \pm \sqrt{(-8)^2 - 4 \times 1 \times 15}}{2 \times 1} \\ \\ &=\dfrac{8 \pm \sqrt{64 - 60}}{2} \\ \\ &=\dfrac{8 \pm \sqrt{4}}{2} \\ \\ &=\dfrac{8 \pm 2}{2} \\ \\ \Rightarrow x&=5 \ \ \& \ \ x= 3 \\ \\ \end{aligned}
when x = 5
y = 10 – 2x
⇒ y = 10 – 2 × 5
⇒ y = 0
when x = 3
y = 10 – 2x
⇒ y = 10 – 2 × 3
⇒ y = 4
so the solution to the system of equation is (x, y) = (5, 0) & (3, 4)
Now x + y = 5 + 0 = 5
x + y = 3 + 4 = 7
So x + y = 5 or 7