Problem 18 - Algebra 2

Question

Solve each equation. Check each solution. $$ \frac{2}{y}+\frac{1}{2}=\frac{5}{2 y} $$

Step-by-Step Solution

Answer

The solution to the equation is \( y = 1 \)

Step 1: Clear the fractions

The first step to solving this fraction-based equation is to eliminate the fractions. A common method of doing this is to multiply every term by the denominator of every fraction in the equation, which is known as the 'least common multiple' (LCM). Here, the LCM is \( 2y \). So, multiply every term in the equation as follows:\( 2y * \frac{2}{y} + 2y * \frac{1}{2} = 2y * \frac{5}{2y} \)That yields: \( 4 + y = 5 \)

Step 2: Solve for \( y \)

Now, solve for \( y \) by bringing the \( y \)-term from right side of the equation to left. We subtract \( y \) from both sides to get:\( 4 + y - y = 5 - y \)That simplifies to: \( 4 = 5 - y \)

Step 3: Final calculation for \( y \)

Finally, solve for \( y \) by subtracting 5 from both sides of the equation and multiply the result by -1 to get \( y \) positive:\( 4 - 5 = 5 - y - 5 \)This simplifies to:\( -1 = - y \) Multiplying by -1 gives:\( y = 1 \)

Step 4: Checking the solution

Substitute \( y = 1 \) back into the original equation to check if the left side equals the right side:\(\frac{2}{1}+\frac{1}{2} = 2 + 0.5 = 2.5\) and \(\frac{5}{2*1} = 2.5 \)Since both sides of the equation are equal, \( y = 1 \) is the correct solution.