Remark.The easiest way to do this question is to graph it. Start with that.
The red line is y = (1/3)^x
The blue line is y = 5*(1/3)^x
CommentThe red line's y intercept is (0,1)
The blue line's y intercept is (0,5)
WhyIf the x value is 0 then (1/3)^0 = 1
y = (1/3)^0 = 1 * 1 = 1 and for the blue graph
y = 5*(1/3)^0 = 5*1 = 5
In other words, in this set of equations, the 5 makes the y intercept 5 times larger than the 1 in front of y = (1/3)^x
If you have choices, could you please list them? I may be giving you the right answer but not in the form required.
Answer:
![\sf -11+7\sqrt{2}](https://tex.z-dn.net/?f=%5Csf%20-11%2B7%5Csqrt%7B2%7D)
Step-by-step explanation:
Given expression:
![\sf \dfrac{3-\sqrt{32}}{1+\sqrt{2} }](https://tex.z-dn.net/?f=%5Csf%20%5Cdfrac%7B3-%5Csqrt%7B32%7D%7D%7B1%2B%5Csqrt%7B2%7D%20%7D)
Rewrite 32 as 16 · 2:
![\sf \implies \dfrac{3-\sqrt{16 \cdot 2}}{1+\sqrt{2} }](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B3-%5Csqrt%7B16%20%5Ccdot%202%7D%7D%7B1%2B%5Csqrt%7B2%7D%20%7D)
Apply radical rule ![\sf \sqrt{a \cdot b}=\sqrt{a}\sqrt{b}](https://tex.z-dn.net/?f=%5Csf%20%5Csqrt%7Ba%20%5Ccdot%20b%7D%3D%5Csqrt%7Ba%7D%5Csqrt%7Bb%7D)
![\sf \implies \dfrac{3-\sqrt{16}\sqrt{2}}{1+\sqrt{2} }](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B3-%5Csqrt%7B16%7D%5Csqrt%7B2%7D%7D%7B1%2B%5Csqrt%7B2%7D%20%7D)
As
:
![\sf \implies \dfrac{3-4\sqrt{2}}{1+\sqrt{2} }](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B3-4%5Csqrt%7B2%7D%7D%7B1%2B%5Csqrt%7B2%7D%20%7D)
Multiply by the conjugate:
![\sf \implies \dfrac{3-4\sqrt{2}}{1+\sqrt{2} } \times \dfrac{1-\sqrt{2} }{1-\sqrt{2} }](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B3-4%5Csqrt%7B2%7D%7D%7B1%2B%5Csqrt%7B2%7D%20%7D%20%5Ctimes%20%5Cdfrac%7B1-%5Csqrt%7B2%7D%20%7D%7B1-%5Csqrt%7B2%7D%20%7D)
![\sf \implies \dfrac{(3-4\sqrt{2})(1-\sqrt{2})}{(1+\sqrt{2})(1-\sqrt{2})}](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B%283-4%5Csqrt%7B2%7D%29%281-%5Csqrt%7B2%7D%29%7D%7B%281%2B%5Csqrt%7B2%7D%29%281-%5Csqrt%7B2%7D%29%7D)
![\sf \implies \dfrac{3-3\sqrt{2}-4\sqrt{2}+4\sqrt{2}\sqrt{2}}{1-\sqrt{2}+\sqrt{2}-\sqrt{2}\sqrt{2}}](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B3-3%5Csqrt%7B2%7D-4%5Csqrt%7B2%7D%2B4%5Csqrt%7B2%7D%5Csqrt%7B2%7D%7D%7B1-%5Csqrt%7B2%7D%2B%5Csqrt%7B2%7D-%5Csqrt%7B2%7D%5Csqrt%7B2%7D%7D)
As
:
![\sf \implies \dfrac{3-3\sqrt{2}-4\sqrt{2}+4 \cdot 2}{1-\sqrt{2}+\sqrt{2}-2}](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B3-3%5Csqrt%7B2%7D-4%5Csqrt%7B2%7D%2B4%20%5Ccdot%202%7D%7B1-%5Csqrt%7B2%7D%2B%5Csqrt%7B2%7D-2%7D)
![\sf \implies \dfrac{3-7\sqrt{2}+8}{1-2}](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B3-7%5Csqrt%7B2%7D%2B8%7D%7B1-2%7D)
![\sf \implies \dfrac{11-7\sqrt{2}}{-1}](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20%5Cdfrac%7B11-7%5Csqrt%7B2%7D%7D%7B-1%7D)
![\sf \implies -11+7\sqrt{2}](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20-11%2B7%5Csqrt%7B2%7D)
Answer:
The Answer is D
Step-by-step explanation:
All you do is add both equations 3^x+10x+2x-4
then you add the like varibles which is 10x+2x=12x
So, the answer is 3^x+12x-4
Answer:
The two points solutions to the system of equations are: (2, 3) and (-1,6)
Step-by-step explanation:
These system of equations consists of a parabola and a line. We need to find the points at which they intersect:
![x^2-2x+3=-x+5\\x^2-2x+x+3-5=0\\x^2-x-2=0\\(x-2)(x+1)=0](https://tex.z-dn.net/?f=x%5E2-2x%2B3%3D-x%2B5%5C%5Cx%5E2-2x%2Bx%2B3-5%3D0%5C%5Cx%5E2-x-2%3D0%5C%5C%28x-2%29%28x%2B1%29%3D0)
Since we were able to factor out the quadratic expression, we can say that the x-values solution of the system are:
x = 2 and x = -1
Now, the associated y values we can get using either of the original equations for the system. We pick to use the linear equation for example:
when x = 2 then ![y=-(2)+5=3](https://tex.z-dn.net/?f=y%3D-%282%29%2B5%3D3)
when x= -1 then ![y=-(-1)+5=6](https://tex.z-dn.net/?f=y%3D-%28-1%29%2B5%3D6)
Then the two points solutions to the system of equations are: (2, 3) and (-1,6)