Question

Part A: Explain why the x-coordinates of the points where the graphs of the equations

Answer 1

Explanation:

Part A. An ordered pair (x, y) is a solution to an equation when subsituting those values into the equation creates a true statement.

The equation 4^(-x) = 8^(-x-1) is the same as 4^(-x) = y = 8^(-x-1), which is to say that the same ordered pair (x, y) satisfies both of the equations ...

• y = 4^(-x)
• y = 8^(-x-1)

If the same ordered pair (x, y) satsifies both equations, (x, y) is on the graph of both equations, so the graphs intersect at that point.

Part B. See the attachment for the tables.

Part C. The equations can be solved graphically by graphing both equations, then identifying the point where they intersect. The attachment shows that point to be (x, y) = (-3, 64). So, the solution to 4^(-x) = 8^(-x-1) is x=-3.

The equations can also be solved graphically by plotting the graph of ...

... 8^(-x-1) - 4^(-x)

and looking for the value of x that makes the difference be zero. Many graphing calculators easily identify x-intercepts, so this can be an easy way to find the solution.

Answer 2

A. Answer: It is a system of equations that can be solved using substitution method; x = -3

Step-by-step explanation:

y = 4⁻ˣ

y = 8⁻ˣ⁻¹

Use substitution to replace y in the second equation with 4⁻ˣ

4⁻ˣ = 8⁻ˣ⁻¹

Solve by manipulating the 4 and 8 so they have the same base, then set the exponents equal to each other and solve for x

2²⁽⁻ˣ⁾ = 2³⁽⁻ˣ⁻¹⁾

2(-x) = 3(-x - 1)

-2x = -3x - 3

  x = -3

Plug the x-value into either of the equations to solve for y

y = 4⁻ˣ

y = 4⁻⁽⁻³⁾

y = 4³

y = 64

Solution is: (-3, 64)

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B.

$\begin{array}{c||c}\ \quad \underline{y=4^{-x}\quad} \ &\quad \underline{y=8^{-x-1}}\quad \\\end{array}\\\begin {array}{c|c||c|c}\quad \underline{x} \quad&\quad \underline{y} \quad &\quad \underline{x} \quad&\quad \underline{y} \quad \\ -3&64&-3&64\\&&&\\-2&16&-2&8\\&&&\\-1&4&-1&1\\&&&\\0&1&0&\dfrac{1}{8}\\ &&&\\1&\dfrac{1}{4}&1&\dfrac{1}{64}\\&&&\\2&\dfrac{1}{16}&2&\dfrac{1}{512}\\&&&\\3&\dfrac{1}{64}&3&\dfrac{1}{4096}\\\end{array}$

Notice that they share the same x, y-coordinate: (-3, 64)

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C. Answer: Graph both equations to see where they cross

Step-by-step explanation:

see attached graph