$3(x+5)-2=10$
Distribute (to get rid of parentheses) | |
Combine any Like Terms | |
Undo addition or subtraction first | |
Undo multiplication or division next | |
We are tasked with solving the linear equation \(3(x+5) - 2 = 10\) for the variable \(x\). The equation includes a constant multiplying a binomial and other constants on both sides. To solve this, we will use algebraic manipulation to isolate \(x\). Our strategy involves distributing the multiplier, simplifying by combining constants, and then using inverse operations to solve for \(x\).
The first step is to distribute the constant \(3\) to each term inside the parentheses:
\[3 \cdot x + 3 \cdot 5 - 2 = 10\]
Calculating the multiplication gives:
\[3x + 15 - 2 = 10\]
Next, we combine the constant terms on the left-hand side of the equation. We have \(15 - 2\), which simplifies to \(13\):
\[3x + 13 = 10\]
To isolate the term with \(x\), we subtract \(13\) from both sides of the equation:
\[3x = 10 - 13\]
\[3x = -3\]
Finally, we divide both sides by \(3\) to solve for \(x\):
\[x = \frac{-3}{3}\]
\[x = -1\]
Thus, the solution to the equation is \(x = -1\).
Algebraic equations are used extensively in various fields, such as finance, engineering, and science. For example, if a store owner knows that after selling 3 units of an item plus an extra 5 items (bundled) and then giving back a 2 dollar discount, the net profit is 10 dollars, they can use an equation like \(3(x+5)-2=10\) to determine the unit price \(x\) of the item. Solving this helps them understand their pricing structure and optimize their business decisions.