Problem 24 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{3(y+3)}{5}=2 y+6 $$

Step-by-Step Solution

Answer

The solution is \(y = -3\).

Step 1: Eliminate the Fraction

To clear the fraction, multiply both sides of the equation by 5 to eliminate the denominator: \[5 \times \frac{3(y+3)}{5} = 5 \times (2y + 6)\]This simplifies to:\[3(y+3) = 10y + 30\]

Step 2: Distribute on Left-hand Side

Distribute the 3 into the parentheses on the left-hand side:\[3y + 9 = 10y + 30\]

Step 3: Move Variables to One Side

Subtract 3y from both sides to get all terms with y on one side:\[3y - 3y + 9 = 10y - 3y + 30\]This simplifies to:\[9 = 7y + 30\]

Step 4: Isolate the Variable

Subtract 30 from both sides to isolate the term with y:\[9 - 30 = 7y + 30 - 30\]This simplifies to:\[-21 = 7y\]

Step 5: Solve for y

Divide both sides by 7 to solve for y:\[\frac{-21}{7} = y\]Which simplifies to:\[y = -3\]

Key Concepts

Eliminating Fractions

When solving linear equations, it's common to encounter fractions. Eliminating fractions simplifies the equation and makes it easier to work with integers. In the given exercise, the equation:\[ \frac{3(y+3)}{5} = 2y + 6 \]contains a fraction on the left-hand side. To remove the fraction, you can multiply every term in the equation by the denominator, which is 5 in this case. This step ensures that the fraction disappears:

• Multiply both sides of the equation by 5: \[ 5 \times \frac{3(y+3)}{5} = 5 \times (2y + 6) \]
• The fraction cancels out, leaving you with: \[ 3(y+3) = 10y + 30 \]

By doing this, the fraction is eliminated, and the equation becomes more straightforward to solve.

Distributive Property

Once fractions are eliminated, the next step often involves using the distributive property. This mathematical property allows us to multiply a single term by a sum or difference within parentheses. In the equation:\[ 3(y+3) = 10y + 30 \]the distributive property can be applied to the term on the left:

• Distribute 3 into \((y+3)\), resulting in: \[ 3y + 9 = 10y + 30 \]

This step expands the terms, removing the parentheses, and enables the rearrangement of terms needed for solving the equation. Remember, applying the distributive property correctly is crucial for maintaining equality between both sides of the equation.

Isolating Variables

With expanded terms, focus shifts to isolating the variable you want to solve for. Isolation means getting the variable by itself on one side of the equation. For our equation:\[ 3y + 9 = 10y + 30 \]the goal is to have all terms involving \(y\) on one side:

• Subtract \(3y\) from both sides to begin gathering \(y\) terms together: \[ 3y - 3y + 9 = 10y - 3y + 30 \]
• This simplifies the equation to: \[ 9 = 7y + 30 \]

Next, continue isolating \(y\) by moving constants away from the \(y\)-term:

• Subtract 30 from both sides to further isolate \(y\): \[ 9 - 30 = 7y + 30 - 30 \]
• Leading to: \[ -21 = 7y \]

The key is to maintain balance by performing the same operation on both sides of the equation at each step.

Simplifying Equations

The final steps of solving linear equations often deal with simplifying to find the value of the variable. With \( -21 = 7y \), the solution can be found by simplifying:

• Divide both sides by 7 to isolate \(y\): \[ \frac{-21}{7} = y \]
• Simplifying gives: \[ y = -3 \]

Simplification makes our solutions cleaner and more comprehensible. Simplifying equations typically involves dividing or multiplying to cancel coefficients of the variable, finally arriving at a solution. This technique ensures clarity and correctness in the answer, revealing that \( y = -3 \) for our equation.