Systems of Exponential Equations and Inequalities
Grade 10 — Algebra
1. Definition
A system of exponential equations contains two or more equations where the unknown appears in the exponent. All equations must be satisfied simultaneously by the same values of the unknowns.
For example, consider the system:
$$\begin{cases} 2^x \cdot 3^y = 72 \\ 2^x + 3^y = 12 \end{cases}$$
Here, both $x$ and $y$ appear as exponents. We must find values of $x$ and $y$ that make both equations true at the same time.
Key Properties of Exponentials
- $a^x > 0$ for all real $x$ (when $a > 0$, $a \neq 1$)
- $a^x = a^y \Leftrightarrow x = y$ (one-to-one property)
- $a^{x+y} = a^x \cdot a^y$
- $a^{x-y} = \dfrac{a^x}{a^y}$
Inequality Direction Rules
- If $a > 1$: $a^x > a^y \Leftrightarrow x > y$ preserve sign
- If $0 < a < 1$: $a^x > a^y \Leftrightarrow x < y$ reverse sign
Interactive: Exponential Functions $y = 2^x$ and $y = 3^x$
Observe how exponential functions are always positive and strictly monotonic. Drag the point to explore values.
Interactive visualization — available in the full lesson