Problem 45 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$2 x+4=-3(x-2)$$

Step-by-Step Solution

Answer

The solution of the equation is \(x = \frac{2}{5}\).

Step 1: Eliminate the brackets

By multiplying -3 inside the brackets on the right side of the equation, the equation changes to: \(2x + 4 = -3x + 6\).

Step 2: Group the like terms

Shifting the variables to the left side and constants to the right side of the equation: \(2x + 3x = 6 - 4\).

Step 3: Simplify the equation

The equation becomes \(5x = 2\).

Step 4: Solve for x

Isolating the variable 'x' by dividing both sides of the equation by 5: \(x = \frac{2}{5}\).

Step 5: Check the solution

Substitute \(x = \frac{2}{5}\) in the original equation: \(2(\frac{2}{5}) + 4 = -3(\frac{2}{5} - 2)\). Simplifying both sides, we get 0 = 0, which is true. Thus, our solution is correct.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when plotted on a graph. They have one or two variables, usually written in the form of \(ax + b = c\). In such expressions:

• \(a\) represents the coefficient of the variable.
• \(b\) is the constant term.
• \(c\) is usually a constant to which the expression is equal.

Solving linear equations involves finding the value of the variable that makes the equation true. In our example, \(2x + 4 = -3(x-2)\), we solve by rearranging terms and performing operations to isolate \(x\). Notice that every term of \(x\) or a constant can be manipulated to maintain balance, which is crucial for solving such equations effectively.
Linear equations are foundational many fields, as they simplify complex relationships into understandable, straightforward expressions.

Algebraic Manipulation

Algebraic manipulation involves using arithmetic operations to simplify or rearrange equations. It is important to follow systematic steps to maintain equation balance and achieve correct results:

• Eliminate brackets: Using distribution, apply any multiplication across terms within brackets.
• Group like terms: Combine terms with similar variables or constants, moving them to respective sides of the equation.
• Simplify the equation: Reduce the equation to its simplest form, often isolating the variable to one side.
• Solve for the variable: Perform necessary operations, like division or subtraction, to find the exact value of the variable.

In our original example, we start by distributing \(-3\) across \((x-2)\), and then rearrange terms to form \(5x = 2\). Finally, dividing both sides by 5 provides the solution \(x = \frac{2}{5}\). This clear sequence of actions is key in algebraic manipulation to accurately solve linear equations.

Checking Solutions

Checking solutions is the process of confirming the correctness of the variable value obtained. This involves substituting the solution back into the original equation and ensuring both sides of the equation are equal.
For example, substitute \(x = \frac{2}{5}\) back into the original equation \(2x + 4 = -3(x-2)\). By calculating both sides:

• Left side: \(2\left(\frac{2}{5}\right) + 4 = \frac{4}{5} + 4 = \frac{24}{5}\) or \(4.8\).
• Right side: \(-3\left(\frac{2}{5} - 2\right) = -3(-\frac{8}{5}) = \frac{24}{5}\) or \(4.8\).

Since both sides are equal, our solution \(x = \frac{2}{5}\) is verified. Checking solutions not only prevents errors but also reinforces understanding of the equation’s structure.