What value of $x$ makes the equation true?
$20+4(3 x-5)+2 x=28$
To find the value of \(x\) that satisfies the equation \(20 + 4(3x - 5) + 2x = 28\), we first analyze the structure of the equation. This is a linear equation in one variable, \(x\). Our strategy is to simplify the expression by applying the distributive property and then isolating the variable \(x\).
First, we apply the distributive property to the term \(4(3x - 5)\). Multiplying 4 by each term inside the parentheses:
\[4 \times (3x) - 4 \times 5 = 12x - 20.\]
Substituting this back into the original equation, we get:
\[20 + (12x - 20) + 2x = 28.\]
Next, we combine the constant terms and the variable terms on the left side:
\[(20 - 20) + (12x + 2x) = 28.\]
Since \(20 - 20 = 0\) and \(12x + 2x = 14x\), the equation simplifies significantly to:
\[14x = 28.\]
Finally, to isolate \(x\), we divide both sides of the equation by 14:
\[x = \frac{28}{14} = 2.\]
Therefore, the value of \(x\) that satisfies the given equation is 2.
Linear equations are foundational in many everyday situations where you need to determine an unknown quantity. For instance, suppose you have a fixed budget for an event (represented by 28 units), and you have a base fee (20 units) and variable costs (4 packs of items, each with 3 units of cost minus a discount of 5, plus 2 extra service units). You can use this linear equation to calculate how many units \(x\) you can afford to manage within your budget. Understanding how to solve such equations allows you to make informed financial or logistical decisions efficiently.