Question

Solve the system y = -x + 7 and y = -0.5(x - 3)^2 + 8.
Please explain your answer in two or more sentences.
(I will give 20 points)

Answer 1

Answer:

(1,6) & (7,0)

Step-by-step explanation:

y = -x + 7

y = -0.5(x - 3)² + 8

To solve the system, solve these two equations simultaneously

-x + 7 = -0.5(x - 3)² + 8

-x + 7 = -0.5(x² - 6x + 9) + 8

-x + 7 = -0.5x² + 3x - 4.5 + 8

0.5x² - 4x + 3.5 = 0

x² - 8x + 7 = 0

x² - 7x - x + 7 = 0

x(x - 7) - (x - 7) = 0

(x - 1)(x - 7) = 0

x = 1, 7

y = -1 + 7 = 6

y = -7 + 7 = 0

(1,6) (7,0)

Since the system has two distinct solutions, the line and the curve meet at two distinct poibts9: (1,6) & (7,0)

Answer 2

Solution of the system is (7,0)  and (1,6)

What is Quadratic equation?

In algebra, a quadratic equation is any equation that can be rearranged in standard form as $ax^{2} +bx+c=0$

Given,

y=-x+7

$y=-0.5(x-3)^{2}+8$

Substitute the value of y in the second equation

$-x+7=-0.5(x^{2} -6x+9)+8\\-x+7=-0.5x^{2} +3x-4.5+8\\-x+7=-0.5x^{2} +3x+3.5\\-0.5x^{2} +4x-3.5=0\\0.5x^{2} -4x+3.5=0$

Multiply both side by 2

$x^{2} -8x+7=0$

split the mid term

$x^{2} -7x-x+7=0\\x(x-7)-(x-7)=0\\(x-7)(x-1)=0$

Therefore value of the x =7 and 1

When x=7

Then y=-7+7=0

when x=1

then y=-1+7=6

Hence, the results of the system is (7,0) and (1,6)

Learn more about quadratic equation here

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