Problem 26 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(\frac{2}{3} y+\frac{1}{2}(y-3)=\frac{y+1}{4}\)

Step-by-Step Solution

Answer

The solution is \( y = \frac{21}{11} \).

Step 1: Distribute inside the parentheses

Expand the equation by distributing the term inside the parentheses. Therefore, the equation transforms as follows: \[ \frac{2}{3}y + \frac{1}{2}(y-3) = \frac{y+1}{4} \] Distribute: \[ \frac{1}{2} \times (y-3) = \frac{1}{2}y - \frac{3}{2} \] So the equation becomes: \[ \frac{2}{3} y + \frac{1}{2} y - \frac{3}{2} = \frac{y+1}{4} \]

Step 2: Combine like terms

Combine the \(y\) terms on the left-hand side:\[ \frac{2}{3} y + \frac{1}{2} y = \frac{4}{6} y + \frac{3}{6} y = \frac{7}{6} y \] Therefore, the equation reduces to: \[ \frac{7}{6} y - \frac{3}{2} = \frac{y+1}{4} \]

Step 3: Eliminate fractions

Multiply every term by 12 (the least common multiple of 6, 2, and 4) to eliminate fractions:\[ 12 \times \left( \frac{7}{6} y - \frac{3}{2} \right) = 12 \times \left( \frac{y+1}{4} \right) \] This results in: \[ 14y - 18 = 3(y+1) \]

Step 4: Expand and simplify

Expand and simplify the right side:\[ 3(y+1) = 3y + 3 \] So the equation becomes:\[ 14y - 18 = 3y + 3 \]

Step 5: Isolate the variable

Subtract \(3y\) from both sides to gather the \(y\) terms on one side: \[ 14y - 3y - 18 = 3 \] This simplifies to: \[ 11y - 18 = 3 \] Then, add 18 to both sides to isolate \(11y\): \[ 11y = 21 \]

Step 6: Solve for \(y\)

Finally, solve for \(y\) by dividing both sides by 11: \[ y = \frac{21}{11} \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental tool in algebra used to simplify expressions. It involves multiplying a single term by each term within a set of parentheses. In this exercise, we apply the Distributive Property to the term \( \frac{1}{2}(y - 3) \). This means you multiply \( \frac{1}{2} \) by both \( y \) and \(-3 \), yielding \( \frac{1}{2}y - \frac{3}{2} \).

• This step is crucial because it helps to eliminate parentheses and simplify the structure of the equation.
• Remember that each term inside the parentheses gets multiplied by the factor outside the parentheses.

By doing this, the equation becomes easier to manipulate in subsequent steps. The proper use of the Distributive Property lays a solid foundation for the rest of the problem.

Combining Like Terms

Combining like terms is an important step in solving equations, as it helps to simplify equations further. This step makes your equation look cleaner and easier to manage. In our exercise, after applying the Distributive Property, the equation is:\[ \frac{2}{3} y + \frac{1}{2}y - \frac{3}{2} = \frac{y+1}{4} \]To combine the terms on the left-hand side effectively:

• Notice both \( \frac{2}{3}y \) and \( \frac{1}{2}y \) are like terms because they both contain the variable \( y \).
• Convert them into a common fraction base and add: \( \frac{2}{3}y + \frac{1}{2}y = \frac{4}{6}y + \frac{3}{6}y = \frac{7}{6}y \).

By doing this, we simplify the left side to \( \frac{7}{6}y - \frac{3}{2} \). Combining like terms efficiently consolidates the equation, making the entire solving process smoother.

Fraction Elimination

Dealing with fractions can be tricky, so it's often helpful to eliminate them to make calculations simpler. This is done by finding the Least Common Multiple (LCM) of the denominators involved. In our problem, the denominators are 6, 2, and 4. The LCM of these numbers is 12.The next step involves multiplying every term in the equation by 12:

• For the term \( \frac{7}{6}y \), multiplying by 12 gives \( 14y \).
• For \( -\frac{3}{2} \), multiplying by 12 gives \( -18 \).
• For \((y+1)/4\), multiplying by 12 gives \( 3(y+1) \).

This makes the equation: \[ 14y - 18 = 3(y + 1) \]Removing fractions once simplifies the equation significantly, enabling easier manipulation and solution.

Linear Equation Simplification

Simplifying a linear equation involves narrowing it down to its simplest form before isolating the variable. The equation after fraction elimination is:\[ 14y - 18 = 3(y + 1) \]To simplify:

• First, expand the right side by distributing 3 over \( y + 1 \) to get \( 3y + 3 \).
• Simplify the equation by aligning like terms: subtract \( 3y \) from both sides to isolate the \( y \) terms.
• This leads to \( 11y - 18 = 3 \). Then, add 18 to both sides to get \( 11y = 21 \).

Finally, you solve for \( y \) by dividing both sides by 11, which gives the solution \( y = \frac{21}{11} \). By carefully simplifying and solving each step, you reduce potential mistakes and better understand manipulation in algebra.