Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$4(2 y+5)=3(5 y-2)$$

Step-by-Step Solution

Answer

The solution is \(y = \frac{26}{7}\).

Step 1: Expand both sides of the equation

First, apply the distributive property to both sides of the equation. Multiply the numbers outside of the parentheses with each term inside the parentheses.Left side: \(4(2y + 5) = 4 imes 2y + 4 imes 5 = 8y + 20\)Right side:\(3(5y - 2) = 3 imes 5y - 3 imes 2 = 15y - 6\)Thus, the equation becomes:\(8y + 20 = 15y - 6\)

Step 2: Move variables to one side

Subtract \(8y\) from both sides to get all the \(y\) terms on one side of the equation.\(8y + 20 - 8y = 15y - 6 - 8y\)This simplifies to:\(20 = 7y - 6\)

Step 3: Isolate the variable

Add \(6\) to both sides to start isolating \(y\).\(20 + 6 = 7y - 6 + 6\)This simplifies to:\(26 = 7y\)Now, divide every term by \(7\) to solve for \(y\):\(\frac{26}{7} = y\)Thus, \(y = \frac{26}{7}\).

Key Concepts

Distributive Property

The distributive property helps in simplifying expressions by eliminating parentheses. It states that multiplying a number by a sum or difference is the same as doing each multiplication separately. This is often expressed as \(a(b+c) = ab + ac\). By distributing the multiplier to each term inside the parentheses, we can break down complicated expressions into simpler parts.
For example, in the original exercise:

• The expression \(4(2y + 5)\) expands to \(8y + 20\).
• Similarly, \(3(5y - 2)\) expands to \(15y - 6\).

Applying the distributive property correctly leads to a new, simplified equation, without changing its value, making it easier to solve.

Solving for a Variable

When solving for a variable, the goal is to find the value of that variable which satisfies the equation. This requires isolating the variable on one side of the equation, typically by performing equivalent operations on both sides.
Let's revisit the exercise: by transforming the expanded equation \(8y + 20 = 15y - 6\), we start by aiming to place all terms involving \(y\) on one side. A systematic approach is key:

• Subtract \(8y\) from both sides to remove it from the left side.
• This results in the equation: \(20 = 7y - 6\).

Through these steps, we've managed to consolidate all the \(y\) terms, thereby simplifying our path to find the value of \(y\).

Isolating Terms

Isolating terms involves rearranging an equation to place a specific variable, or term, on one side of the equation by itself. It is crucial for uncovering the value of unknowns.
Let's explore the continued solution from our exercise:

• After simplifying to get \(20 = 7y - 6\), add \(6\) to each side to eliminate the constant \(-6\) adjacent to \(7y\).
• This addition results in \(26 = 7y\).

With only one term now on the right, the variable \(y\) is more prominently displayed, making it straightforward to solve for it in upcoming steps.

Equation Balancing

Equation balancing maintains the equality of an equation by performing the same operation on both sides. This principle ensures that while rearranging terms or solving for variables, the equation's balance remains intact.
In our ongoing example, when we reach the simplified form \(26 = 7y\), we divide both sides by \(7\), ensuring the operation doesn’t disrupt the balance:

• Doing so yields \(y = \frac{26}{7}\).

Dividing both sides by \(7\) effectively isolates \(y\), without altering the equality, and it displays the required value of \(y\), completing the solution. This operation highlights the power of balance in preserving the integrity of the mathematical relationships within an equation.