91Ó°ÊÓ

Algebraic expressions are combinations of variables, numbers, and operations. In our example, we work with the expression \(3(x+15)+4(11-x)\). Understanding the structure of an algebraic expression is crucial to solving equations. The variables represent unknown quantities, and it's our job to figure out what they are. To handle these expressions correctly, we follow the order of operations: parentheses first, then multiplication or division, and finally addition or subtraction.

When faced with solving an equation like the one in our example, we're essentially doing detective work to find the value of \(x\) that makes the equation true. Each step taken is like following a clue to uncover the unknown. By methodically expanding brackets and combining like terms, we uncover the simplicity hidden within what initially looks complex.

Combining like terms is a way to simplify algebraic expressions and equations. It's like tidying up a cluttered room by putting similar items together. In the expanded form \(3x + 45 - 4x + 44\), we notice that \(3x\) and \(-4x\) are like terms because they both contain the variable \(x\) raised to the same power. Just as we'd stack books together on a shelf, we combine these like terms by adding or subtracting their coefficients.

Doing this in our example, we get \(-x + 89 = 0\). It's a simpler form, which brings us a step closer to finding our answer. This methodical consolidation is key to making equations more approachable and easier to solve. The goal is always to reduce the equation down to the simplest form possible, transforming what may appear overwhelming into something much more manageable.

Once we believe we've solved an equation, it's tempting to immediately move on, but checking our solution is a vital step. Imagine you've just solved a puzzle; you double-check to ensure every piece is in its place. Similarly, we need to confirm that the value we found for \(x\) really does make the original equation balance.

To check our solution of \(x = 89\), we plug 89 back into the original equation in place of \(x\) and see if we arrive at a true statement. If both sides equal, our solution is correct. It's gratifying when our calculated value fits perfectly, much like the last piece of a jigsaw puzzle. This step doesn't just confirm our solution; it also reinforces our understanding of the relationship between algebraic expressions and their solutions.