Question

Solve the simultaneous equations.
You must show all your working.
y = 4-X
x² + 2y2 = 67
x=
y=
X=

Answer 1

Answer:

(7, - 3 ) and (- $\frac{5}{3}$, $\frac{17}{3}$ )

Step-by-step explanation:

Given the 2 equations

y = 4 - x → (1)

x² + 2y² = 67 → (2)

Substitute y = 4 - x into (2)

x² + 2(4 - x)² = 67 ← expand and simplify left side

x² + 2(16 - 8x + x²) = 67

x² + 32- 16x + 2x² = 67

3x² - 16x + 32 = 67 ( subtract 67 from both sides )

3x² - 16x - 35 = 0 ← in standard form

(x - 7)(3x + 5) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 7 = 0 ⇒ x = 7

3x + 5 = 0 ⇒ 3x = - 5 ⇒ x = - $\frac{5}{3}$

Substitute these values into (1) for corresponding values of y

x = 7 : y = 4 - 7 = - 3 ⇒ (7, - 3 )

x = - $\frac{5}{3}$ : y = 4 + $\frac{5}{3}$ = $\frac{17}{3}$ ⇒ (- $\frac{5}{3}$, $\frac{17}{3}$ )

solutions are (7, - 3 ) and (- $\frac{5}{3}$, $\frac{17}{3}$ )

Answer 2

The solution of the equations are $x=7$ and $x=-1.67$

System of equations:

The given system of equations are,

                 $y=4-x\\\\x^{2} +2y^{2} =67$

Substitute value of y in second equation;

           $x^{2} +2(4-x)^{2} =67\\\\x^{2} +2(16+x^{2} -8x)=67\\\\x^{2} +32+2x^{2} -16x=67\\\\3x^{2} -16x-35=0\\\\x=\frac{16\pm\sqrt{256+420} }{6}\\ \\x=7, -1.67$

The solution of the equations are $x=7$ and $x=-1.67$

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