Solve for $x$.
$-2\left(x+\frac{1}{2}\right)+9 x=20$
$x=$
The problem is to solve the linear equation \(-2(x+\frac{1}{2})+9x=20\) for the variable \(x\). This equation involves distribution, combining like terms, and basic arithmetic operations to isolate the unknown variable. Our goal is to determine the value of \(x\) that satisfies this equality.
First, we distribute the \(-2\) into the parentheses. Applying the distributive property, we get:
\[-2 \cdot x - 2 \cdot \left(\frac{1}{2}\right) + 9x = 20\]
\[-2x - 1 + 9x = 20\]
Now, we group the terms containing \(x\):
\[(-2 + 9)x - 1 = 20\]
\[7x - 1 = 20\]
Next, we isolate the term with \(x\) by adding 1 to both sides of the equation:
\[7x - 1 + 1 = 20 + 1\]
\[7x = 21\]
Finally, to solve for \(x\), we divide both sides by 7:
\[x = \frac{21}{7}\]
\[x = 3\]
Linear equations are essential in everyday situations. For instance, if you are planning a budget for a party and you know you have a fixed cost for room rental plus a variable cost per guest, you can use a linear equation to determine the maximum number of guests you can invite. If the room costs \(\$50\) and each food serving costs \(\$15\), and your budget is \(\$200\), you solve \(15x + 50 = 200\). This helps you calculate how many guests you can accommodate while staying within your financial limits.