Problem 10 - Algebra for College Students

Question

Solve each equation. \(5 x+1=12\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{11}{5}\).

Step 1: Move the Constant

To solve for the variable \(x\), we must first isolate the terms that contain \(x\). Currently, the equation is \(5x + 1 = 12\). To do this, subtract \(1\) from both sides of the equation:\[5x + 1 - 1 = 12 - 1\]This simplifies to:\[5x = 11\]

Step 2: Isolate the Variable

Now that we have \(5x = 11\), divide both sides of the equation by \(5\) to solve for \(x\):\[\frac{5x}{5} = \frac{11}{5}\]This simplifies to:\[x = \frac{11}{5}\]

Step 3: Simplify the Solution

The solution to the equation is currently \(x = \frac{11}{5}\). This fraction cannot be simplified further, so \(x = \frac{11}{5}\) remains our final solution.

Key Concepts

Solving Linear Equations

When faced with a linear equation, the ultimate goal is to find the value of the variable that makes the equation true. Linear equations, like the one in this problem, typically take the form of: \(ax + b = c\). In this format, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we are trying to solve for.

Solving linear equations involves several steps that help simplify and balance the equation, ensuring that the equality remains when adjusting either side. These steps include:

• Eliminating constants from one side
• Ensuring all variable terms are on one side of the equation
• Solving for the variable by dividing or multiplying as needed

By following these steps, we can effectively solve the equation for the unknown variable.

Algebraic Manipulation

Algebraic manipulation is all about re-arranging and simplifying an equation to solve for the unknown. This process involves performing the same arithmetic operation on both sides of an equation to keep it balanced. In our exercise, we start with the equation \(5x + 1 = 12\).

Our first step involves subtracting \(1\) from both sides. This is algebraic manipulation where we ensure the balance of the equation remains intact while simplifying it to bring us one step closer to isolating the variable \(x\). This produces: \[5x = 11\]

Remember, every manipulation aims to maintain equality. Maintaining balance is crucial as imbalance would result in incorrect solutions.

Variable Isolation

Variable isolation means getting our desired variable, here \(x\), by itself on one side of the equation. This is achieved by removing other terms from one side of the equation.

In our example, after simplifying to \(5x = 11\), the next step to isolate \(x\) is dividing both sides of the equation by \(5\). This is direct and straightforward because our goal is to have \(x\) by itself.

As you perform this operation, you get:

• Divide each term by the coefficient of \(x\) (which is \(5\))
• The equation becomes: \[x = \frac{11}{5}\]

By isolating \(x\), we are left with the solution to the equation. This fraction does not simplify further, confirming \(x = \frac{11}{5}\) as our final answer.