Question

solve the following equations. homework help
a. \\(\frac{x}{4} - \frac{x}{6} = \frac{2}{3}\\) b. \\(5^{3x} = 25^{(x + 2)}\\)
which pairs of triangles below are congruent? if the pair is congruent, justify your conclusion by drawing a flowchart an

Explanation:

Part a: Solve \(\boldsymbol{\frac{x}{4}-\frac{x}{6}=\frac{2}{3}}\)

Step1: Find common denominator

The common denominator of 4 and 6 is 12. Rewrite each fraction:
\(\frac{x}{4}=\frac{3x}{12}\), \(\frac{x}{6}=\frac{2x}{12}\), \(\frac{2}{3}=\frac{8}{12}\).
Equation becomes: \(\frac{3x}{12}-\frac{2x}{12}=\frac{8}{12}\).

Step2: Simplify left side

Subtract numerators: \(\frac{3x - 2x}{12}=\frac{8}{12}\) → \(\frac{x}{12}=\frac{8}{12}\).

Step3: Solve for \(x\)

Multiply both sides by 12: \(x = 8\).

Part b: Solve \(\boldsymbol{5^{3x}=25^{(x + 2)}}\)

Step1: Express 25 as \(5^2\)

Rewrite \(25^{(x + 2)}\) as \((5^2)^{(x + 2)}\).

Step2: Apply exponent rule \((a^m)^n=a^{mn}\)

\((5^2)^{(x + 2)} = 5^{2(x + 2)} = 5^{2x + 4}\).

Step3: Set exponents equal (since bases are equal)

\(3x = 2x + 4\).

Step4: Solve for \(x\)

Subtract \(2x\) from both sides: \(x = 4\).

Answer:

s:
a. \(x = 8\)
b. \(x = 4\)