- Distribute the constant into the parentheses: \(4x - 4 + 3 = 11\).
- Combine the constant terms: \(4x - 1 = 11\).
- Add 1 to both sides to isolate the \(x\) term: \(4x = 12\).
- Divide by 4 to solve for \(x\): \(\boxed{3}\).
Explanation
- Analysis of the Problem
We begin by analyzing the given linear equation: \[4(x-1)+3=11.\] Our objective is to solve for the variable \(x\) using a systematic approach that involves distribution, combining like terms, and isolation of the variable. This is a standard procedure for solving first-degree equations.
- Distribution and Combining Like Terms
First, we apply the distributive property to the expression \(4(x-1)\). Multiplying 4 by both terms inside the parentheses gives: \[4x - 4 + 3 = 11.\] Now, we identify the constant terms on the left side, which are \(-4\) and \(+3\). Combining these like terms results in: \[4x - 1 = 11.\]
- Isolating the Variable Term
Next, we need to isolate the term containing the variable, \(4x\). We do this by reversing the subtraction of 1. By adding 1 to both sides of the equation, we obtain: \[4x = 11 + 1,\] which simplifies to: \[4x = 12.\]
- Solving for x
Finally, to solve for \(x\), we must undo the multiplication by 4. Dividing both sides of the equation by 4 gives: \[x = \frac{12}{4}.\] Simplifying this fraction, we arrive at the final solution: \[x = 3.\] We can verify this by plugging \(x=3\) back into the original equation: \(4(3-1)+3 = 4(2)+3 = 8+3 = 11\), which confirms our result is correct.
Examples
Linear equations like this are frequently used in everyday finance. For example, if you hire a plumber who charges a flat fee of \(\$3\) for a service visit plus \(\$4\) for every hour of work, the total cost for \(x\) hours would be modeled by the expression \(4x + 3\). If you were charged \(\$11\) in total, you could solve the equation \(4(x-1) + 3 = 11\) if the flat fee was applied differently or simply \(4x + 3 = 11\) to find out how many hours they worked. In our case, the equation demonstrates how to break down costs step-by-step to find the unknown duration of the service.