Question

that make each system true. be sure to show your work or explain your thinking
b. $18 = a cdot b^2$
$54 = a cdot b^3$
hint (b):
review what the equal value method is, and remember to isolate $a$ first.
answer (b):
$a = 2$
$b = 3$

Explanation:

Step1: Isolate \( a \) from the first equation

From \( 18 = a \cdot b^2 \), we can express \( a \) as \( a=\frac{18}{b^2} \).

Step2: Substitute \( a \) into the second equation

Substitute \( a = \frac{18}{b^2} \) into \( 54 = a \cdot b^3 \). We get \( 54=\frac{18}{b^2}\cdot b^3 \). Simplify the right - hand side: \( \frac{18}{b^2}\cdot b^3 = 18b \). So the equation becomes \( 54 = 18b \).

Step3: Solve for \( b \)

Divide both sides of \( 54 = 18b \) by 18: \( b=\frac{54}{18}=3 \).

Step4: Solve for \( a \)

Substitute \( b = 3 \) into \( a=\frac{18}{b^2} \). Then \( a=\frac{18}{3^2}=\frac{18}{9} = 2 \).

Answer:

\( a = 2 \), \( b = 3 \)