Practice Test 39 - Algebra 1

20 questions. Answer them in any order, then submit to see your score and every explanation.

Solve the system (you may use any method): $\begin{cases} 2x + y = 8 \\ x - y = 1 \end{cases}$ What is the solution $(x, y)$ that satisfies both equations?
Is $(3, 2)$ a solution to the system? $\begin{cases} x + y = 5 \\ 2x - y = 4 \end{cases}$
Why is $5+\sqrt{3}$ irrational? Choose the option with valid reasoning.
Which situation involves a constant percent rate of change (exponential), rather than a constant additive change (linear)?
Which reasoning best justifies the definition $27^{2/3}=(\sqrt[3]{27})^2$ rather than treating $\tfrac{2}{3}$ as ordinary division (assume $b>0$ )?
What are the solutions to the equation $x^2 - 9 = 0$ ? (Use factoring.)
The expression $3^{2x}$ represents the area of a square region after $x$ time periods. Which equivalent form reveals the length of one side of the square?
The temperature $T$ (in degrees Celsius) of a cooling object after $t$ minutes follows the equation $T = 80 \cdot (0.95)^t + 20$ . Which equation would you solve to find when the temperature drops to exactly 35°C?
Which statement is the best general explanation for why the sum of two rational numbers is always rational? (Recall: a rational number can be written as $\frac{p}{q}$ where $p,q$ are integers and $q\ne 0$ .)
An investment is modeled by $A(t)=5000(1.02)^t$ , where $t$ is in years. What is the value of the growth factor $b$ in $A=a\cdot b^t$ ?
Use proof by contradiction to justify the correct classification: Why must $5+\sqrt{3}$ be irrational?
A student claims: " $16^{1/4}$ should mean the fourth root of 16." Which reasoning best supports this definition using exponent properties (assume $b>0$ )?
Two competing technologies are being adopted according to these models: Technology A follows $A(t) = 1000t^2 + 500t$ , and Technology B follows $B(t) = 100 \cdot 3^t$ , where $t$ represents months after launch. A market analyst notes that at $t = 4$ months, Technology A has 18,000 users while Technology B has approximately 8,100 users. She predicts that Technology A will maintain its lead indefinitely because "quadratic growth is more powerful than exponential growth with small bases." Evaluate her prediction.
The function $f(x) = 2x^2 - 12x + 25$ represents the temperature above ground level at height $x$ meters. At what height is the temperature minimum, and what is that minimum temperature?
A geometric sequence has $g_3 = 12$ and $g_6 = 96$ . What is the recursive formula for this sequence, and what is the value of $g_1$ ?
A linear function $f(x) = 5x + 10$ and an exponential function $g(x) = 2^x$ are compared for positive integer values of $x$ . At $x = 4$ , we have $f(4) = 30$ and $g(4) = 16$ , so the linear function has a greater value. What can be concluded about the relationship between these functions for large values of $x$ ?
The formula for electrical power is $P = I^2R$ , where $P$ is power, $I$ is current, and $R$ is resistance. An electrical engineer needs to determine the current when the power is quadrupled and the resistance is doubled. If the original current was $I_0$ , what is the new current?
If $f(x)=5-x$ and $g(x)=3x$ , what is $g(f(x))$ ?
A neighborhood group wants to describe traffic on one street during the 30 minutes after school ends. Which set of quantities is most appropriate for a descriptive model of current traffic flow?
The sum of two numbers is 15 and their difference is 3. Let $x$ be the larger number and $y$ be the smaller number. What are $(x,y)$ ?

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