Question

Solve the equation \(2[(2 / 3) \mathrm{y}+5]+2(\mathrm{y}+5)=130\)

Short answer

The solution to the equation \(2[(2 / 3)y + 5] + 2(y + 5) = 130\) is \(y = 33\).

Step-by-step solution

Distribute the numbers outside the brackets

To distribute the 2 outside each bracket, multiply each term inside the brackets by 2: \(2[(2 / 3) y + 5] + 2(y + 5) = 130\) \(2*(\frac{2}{3}y)+2*5+2y+2*5=130\)

Simplify the equation by combining like terms

We will now simplify the equation by combining like terms: \(\frac{4}{3}y + 10 + 2y + 10 = 130\) Combine the y terms: \(\frac{4}{3}y + 2y\) To add these terms, find a common denominator, which is 3. Write 2y as \(\frac{6}{3}y\): \(\frac{4}{3}y + \frac{6}{3}y = \frac{10}{3}y\) Now, the equation becomes: \(\frac{10}{3}y + 20 = 130\)

Isolate the variable y on one side of the equation

Subtract 20 from both sides of the equation: \(\frac{10}{3}y = 110\)

Solve for y

To solve for y, divide both sides of the equation by \(\frac{10}{3}\). An alternative method is to multiply both sides by the reciprocal, which is \(\frac{3}{10}\): \(y = \frac{3}{10} * 110\) Now, simplify and calculate the value of y: \(y = 33\) So, the solution to the equation \(2[(2 / 3) y + 5] + 2(y + 5) = 130\) is \(y = 33\).

Key concepts

Algebraic Manipulation

Algebraic manipulation is a core skill in mathematics that allows you to rearrange and simplify equations to solve for unknowns. In the original exercise, algebraic manipulation begins with distributing the constant outside the bracket. This involves multiplying each term inside the brackets by 2. For instance, distributing between brackets involves calculating

• firstly, the term \(2(\frac{2}{3}y + 5)\)
• and then \(2(y + 5)\)

After distributing, the equation becomes \(2 \cdot \frac{2}{3}y + 2 \cdot 5 + 2y + 2 \cdot 5 = 130\).
The principle behind this step is rooted in the distributive property of multiplication over addition, allowing us to manipulate algebraic expressions and prepare them for further simplification. By handling one operation at a time, it becomes easier to focus on each particular part of the equation, making sure that no term is left unattended.
Through practice, algebraic manipulation helps to recognize patterns and develop problem-solving skills critical to algebra.

Equation Simplification

Equation simplification is a process that helps make equations easier to understand and solve. After distributing,
the next goal in our solution is to simplify the equation further by combining like terms. It involves reducing the number of terms without changing the equation's value.
This makes it simpler and easier to work with.

• This is evident as the expression: \(\frac{4}{3}y + 10 + 2y + 10 = 130\) is simplified.
• Adding similar terms like \(10 + 10\) reduces these to \(20\).

This stage works harmoniously with algebraic manipulation and sets the foundation for solving the equation. It also provides clarity by removing unnecessary parts and displaying the mathematical structure clearly.
Without simplification, equations would remain long and cumbersome, making them difficult to interpret and solve.

Combining Like Terms

Combining like terms is critical, especially when dealing with multiple terms of the same variable. It involves adding or subtracting coefficients, making the equation more straightforward to solve.
In the given equation, after distributing, you have terms like \(\frac{4}{3}y\) and \(2y\) that can be combined:
To do this, you have to think of equations in terms of fractions. Consider \(2y\) as \(\frac{6}{3}y\).

• The combination of \(\frac{4}{3}y + \frac{6}{3}y\) becomes \(\frac{10}{3}y\).
• Converting unlike terms to a common denominator, like 3 in this case, facilitates their combination.

This results in the compact expression \(\frac{10}{3}y + 20 = 130\).
Not only does it simplify the physical structure of the equation, but it also allows for better focus on solving for the variable \(y\).
The ability to combine like terms stems from understanding how coefficients relate to each other, allowing you to see patterns and solutions faster.