The difference between Tucker and Karly's take is that Tucker's solution is analytical while Karly's is graphical. But both are correct either way.
For Tucker's solution, let's say at x=-3 the value for y is 4, and at x=3, the value of y is still 4, then the average rate of change or slope is 0. Note that the slope of the curve is Δy/Δx. Since there is no change for Δy, the slope is zero.
For Karly's solution, even if the curve travels high or low but would have the same elevation of x=-3 and x=3, the average rate of change is still zero. It is actually just same with Tucker's but Karly just verbalizes her solution that was observed visually.
Answer:
![(x,y)=\left(\; \boxed{-1,-8} \; \right)\quad \textsf{(smaller $x$-value)}](https://tex.z-dn.net/?f=%28x%2Cy%29%3D%5Cleft%28%5C%3B%20%5Cboxed%7B-1%2C-8%7D%20%5C%3B%20%5Cright%29%5Cquad%20%5Ctextsf%7B%28smaller%20%24x%24-value%29%7D)
![(x,y)=\left(\; \boxed{5,16} \; \right)\quad \textsf{(larger $x$-value)}](https://tex.z-dn.net/?f=%28x%2Cy%29%3D%5Cleft%28%5C%3B%20%5Cboxed%7B5%2C16%7D%20%5C%3B%20%5Cright%29%5Cquad%20%5Ctextsf%7B%28larger%20%24x%24-value%29%7D)
Step-by-step explanation:
Given system of equations:
![\begin{cases}y=x^2-9\\y=4x-4\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dy%3Dx%5E2-9%5C%5Cy%3D4x-4%5Cend%7Bcases%7D)
To solve by the method of substitution, substitute the first equation into the second equation and rearrange so that the equation equals zero:
![\begin{aligned}x^2-9&=4x-4\\x^2-4x-9&=-4\\x^2-4x-5&=0\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dx%5E2-9%26%3D4x-4%5C%5Cx%5E2-4x-9%26%3D-4%5C%5Cx%5E2-4x-5%26%3D0%5Cend%7Baligned%7D)
Factor the quadratic:
![\begin{aligned}x^2-4x-5&=0\\x^2-5x+x-5&=0\\x(x-5)+1(x-5)&=0\\(x+1)(x-5)&=0\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dx%5E2-4x-5%26%3D0%5C%5Cx%5E2-5x%2Bx-5%26%3D0%5C%5Cx%28x-5%29%2B1%28x-5%29%26%3D0%5C%5C%28x%2B1%29%28x-5%29%26%3D0%5Cend%7Baligned%7D)
Apply the <u>zero-product property</u> and solve for x:
![\implies x+1=0 \implies x=-1](https://tex.z-dn.net/?f=%5Cimplies%20x%2B1%3D0%20%5Cimplies%20x%3D-1)
![\implies x-5=0 \implies x=5](https://tex.z-dn.net/?f=%5Cimplies%20x-5%3D0%20%5Cimplies%20x%3D5)
Substitute the found values of x into the <u>second equation</u> and solve for y:
![\begin{aligned}x=-1 \implies y&=4(-1)-4\\y&=-4-4\\y&=-8\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dx%3D-1%20%5Cimplies%20y%26%3D4%28-1%29-4%5C%5Cy%26%3D-4-4%5C%5Cy%26%3D-8%5Cend%7Baligned%7D)
![\begin{aligned}x=5 \implies y&=4(5)-4\\y&=20-4\\y&=16\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dx%3D5%20%5Cimplies%20y%26%3D4%285%29-4%5C%5Cy%26%3D20-4%5C%5Cy%26%3D16%5Cend%7Baligned%7D)
Therefore, the solutions are:
![(x,y)=\left(\; \boxed{-1,-8} \; \right)\quad \textsf{(smaller $x$-value)}](https://tex.z-dn.net/?f=%28x%2Cy%29%3D%5Cleft%28%5C%3B%20%5Cboxed%7B-1%2C-8%7D%20%5C%3B%20%5Cright%29%5Cquad%20%5Ctextsf%7B%28smaller%20%24x%24-value%29%7D)
![(x,y)=\left(\; \boxed{5,16} \; \right)\quad \textsf{(larger $x$-value)}](https://tex.z-dn.net/?f=%28x%2Cy%29%3D%5Cleft%28%5C%3B%20%5Cboxed%7B5%2C16%7D%20%5C%3B%20%5Cright%29%5Cquad%20%5Ctextsf%7B%28larger%20%24x%24-value%29%7D)
The 2 angles add up to 90 degrees so
x + y = 90, also:-
x - y = 72 (given)
adding the 2 equations:-
2x = 162
x = 81
and therefore
y + 81 = 90
y = 9
Answer the 2 angles are 81 and 9 degrees
Answer:
To determine whether a decimal is rational or not, you need to know that...
Irrational numbers don't end and have no pattern whereas rational numbers are the complete opposite. Rational numbers end and have a repeating pattern.
Step-by-step explanation:
Here are examples of irrational numbers:
0.9384903204..... , π , √2
Examples of rational numbers:
0.777777... (is rational because it has a repeating pattern of 7) , √49
Hope this helps :)