Question

tenisha solved the equation below by graphing a system of equations.

\\\log_{3} 5x = \log_{5} (2x + 8)\\

which point approximates the solution for tenishas system of equations?

(0.9, 0.8)
(1.0, 1.4)
(2.3, 1.1)
(2.7, 13.3)

Explanation:

Define the system of equations

Using the Solving Exponential Equations Graphically knowledge point

$$
LATEXBLOCK0
$$

Evaluate the functions at the given x-values

We will test the \(x\)-coordinates from the options to find which one yields approximately equal \(y\)-values for both equations.

• For \(x = 0.9\):

$$
LATEXBLOCK1
$$

These values are close to each other and approximate the \(y\)-value of \(1.4\) given in the option \((1.0, 1.4)\). Let's check \(x = 1.0\).

• For \(x = 1.0\):

$$
LATEXBLOCK2
$$

At \(x = 1.0\), the \(y\)-values are extremely close (\(1.46 \approx 1.43\)), with both rounding to approximately \(1.4\).

• For \(x = 2.3\):

$$
LATEXBLOCK3
$$

These are not close.

Identify the correct approximating point

Comparing our evaluations, the point \((1.0, 1.4)\) is the best approximation for the intersection of the two curves.

Answer:

• (A) (0.9, 0.8)
• (B) (1.0, 1.4) (Correct answer)
• (C) (2.3, 1.1)
• (D) (2.7, 13.3)