Q1.
y - 3x = 2
y = (x + 1)² - 5
Using the substitution method, we will replace y in the first equation and solve it:
(x + 1)² - 5 - 3x = 2
Since (a + b)² = a² + 2ab² + b², then (x + 1)² = x² + 2·1·x + 1² = x² + 2x + 1
So, let's get back to the equation and replace (x + 1)²:
x² + 2x + 1 - 5 - 3x = 2
x² + 2x - 3x + 1 - 5 - 2 =0
x² - x - 6 = 0
Now use the quadratic formula: ax² + bx + c = 0, [tex]x = \frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} [/tex]
In the equation x² - x - 6 = 0, a = 1, b = -1, c = -6
Thus:
[tex]\frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} =\frac{-(-1)+/- \sqrt{ (-1)^{2}-4*1*(-6) } }{2*1} =\frac{1+/- \sqrt{ 1-(-24)} }{2} [/tex][tex]=\frac{1+/- \sqrt{ 25} }{2} = \frac{1+/-5}{2} [/tex]
Thus, the solutions are:
[tex]x= \frac{1+5}{2} = \frac{6}{2} =3[/tex] ⇒ [tex]y = (x+1)^{2}-5=(3+1)^{2} -5=4^{2} -5=16-5=11[/tex]
[tex]x= \frac{1-5}{2} = \frac{-4}{2}=-2 [/tex] ⇒ [tex]y=(x+1) ^{2} -5=(-2+1)^{2} -5=(-1)^{2} -5=1-5=-4[/tex]
x =3, y = 11 and x = -2, y = -4
Q2.
We have two equations:
y - 3x = 2
y = (x + 1)² - 5
____________
y - 3x = 2
y - (x + 1)² = -5
____________
Let's now, multiply the second equation by (-1):
y - 3x = 2
(-1)·y - (-1)·(x + 1)² = (-1)(-5)
____________
y - 3x = 2
-y + (x + 1)² = 5
____________
Now, add this two equations:
y - 3x + (-y + (x + 1)²) = 2 + 5
y - 3x - y + (x + 1)² = 7
y can be cancelled:
- 3x + (x + 1)² = 7
- 3x + x² + 2x + 1 = 7
x² - x + 1 - 7 = 0
⇒ x² - x - 6 = 0
Now, as in the previous part, use the quadratic formula: ax² + bx + c = 0, [tex]x = \frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} [/tex]
In the equation x² - x - 6 = 0, a = 1, b = -1, c = -6
Thus:
[tex]\frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} =\frac{-(-1)+/- \sqrt{ (-1)^{2}-4*1*(-6) } }{2*1} =\frac{1+/- \sqrt{ 1-(-24)} }{2} [/tex][tex]=\frac{1+/- \sqrt{ 25} }{2} = \frac{1+/-5}{2} [/tex]
Thus, the solutions are:
[tex]x= \frac{1+5}{2} = \frac{6}{2} =3[/tex] ⇒ [tex]y = (x+1)^{2}-5=(3+1)^{2} -5=4^{2} -5=16-5=11[/tex]
[tex]x= \frac{1-5}{2} = \frac{-4}{2}=-2 [/tex] ⇒ [tex]y=(x+1) ^{2} -5=(-2+1)^{2} -5=(-1)^{2} -5=1-5=-4[/tex]
x =3, y = 11 and x = -2, y = -4
