Create your own expression that, when simplified, will be equivalent to $5 y+4$.
To create an expression equivalent to \(5y + 4\), we can use algebraic properties such as the distributive law and the combining of like terms. We want to construct a multi-step expression that, when simplified, results in the target linear expression. A systematic way to do this is to decompose the terms \(5y\) and \(4\) into smaller components and group them using parentheses and additional operations.
We propose the expression:
\[3(y + 2) + 2y - 2\]
To verify this, we first apply the distributive property to the first term \(3(y + 2)\):
\[3(y + 2) = 3y + 3 \times 2 = 3y + 6\]
Substituting this back into the expression, we get:
\[(3y + 6) + 2y - 2\]
Next, we combine the like terms. We group the terms containing \(y\) and the constant terms:
\[(3y + 2y) + (6 - 2)\]
Performing the additions and subtractions, we have:
\[5y + 4\]
The expression simplifies correctly to our target value.
Creating algebraic expressions is like building with blocks. If you want to arrive at a final shape (the target expression), you can combine several smaller blocks (simpler operations) in different ways. In engineering, you might use complex formulas to describe the stress on a beam, but you can often simplify those formulas into a simpler model to make quick, informed decisions on-site without needing a computer to calculate the full equation every time. Understanding how to build and simplify these expressions is a foundational skill in both mathematics and practical problem-solving.