Problem 8 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solution in the original equation. $$ -5 x=3 $$

Step-by-Step Solution

Answer

The solution to the equation \(-5x = 3\) is \(x = -\frac{3}{5}\)

Step 1: Identify the Equation Structure

The initial equation is of the form \(ax = b\) where \(a\) is -5 and \(b\) is 3.

Step 2: Inverse the Operation

The operation (-5 multiplication) applied to \(x\) can be inverted by dividing both sides of the equation by -5. Doing so will give us \(x = \frac{3}{-5}\).

Step 3: Simplify the Result

Simplifying the result will yield \(x = -\frac{3}{5}\).

Step 4: Check the Solution

By substituting the obtained result in the original equation, we have \(-5* -\frac{3}{5} = 3\). The equation checks out correctly, thus \(x = -\frac{3}{5}\) is the solution to the equation.

Key Concepts

Inverse Operations

When solving linear equations, inverse operations help us isolate the variable we are solving for. Essentially, they allow us to 'undo' operations to make manipulation possible. In the given equation \(-5x = 3\), the multiplication of \(x\) by \(-5\) is the operation that needs to be inverted.

• Since \(x\) is multiplied by \(-5\), we apply the inverse operation of multiplication, which is division.
• By dividing both sides of the equation by \(-5\), what happens mathematically is that the \(-5\) on the left-hand side cancels out.
• This leaves us with \(x\), isolated on one side of the equation.

The equation then becomes \(x = \frac{3}{-5}\), illustrating how inverse operations work to solve equations.

Equation Structure

Understanding the structure of a linear equation like \(ax = b\) is critical in solving them. Each component of the equation has a role in guiding how we solve it.

• The \(a\) in the equation \(ax = b\) is the coefficient of \(x\). This tells us what operation is being applied to \(x\). In our example, \(a\) is \(-5\), meaning \(x\) is being multiplied by \(-5\).
• The \(b\) is the constant term on the other side of the equation. For our problem, \(b\) is 3, representing the value that \(ax\) is equal to.

Knowing these parts helps us determine the necessary steps, like identifying that \(-5\) needs to be divided out to isolate \(x\). Correctly understanding and identifying equation structure is a skill that becomes more intuitive with practice.

Solution Checking

Once you obtain a solution, it is crucial to verify its correctness by substituting it back into the original equation. This step ensures that no errors were made during calculations.

• Substitute the solution \(x = -\frac{3}{5}\) back into the original equation, replacing \(x\) with \(-\frac{3}{5}\).
• Calculating \(-5 \times -\frac{3}{5} = 3\), you obtain the initial right-hand side value, 3, which confirms accuracy.
• If both sides balance, the solution is correct.

Solution checking acts as a safety net to catch mistakes that could have occurred in earlier steps. Ensuring accuracy is key to solving equations reliably.

Simplification

Simplification is the process of reducing expressions to their simplest form. In the context of solving equations, this means making sure the solution is expressed in the simplest terms.

• After applying inverse operations, the solution is often a fraction or a simple expression. In our problem, \(x = \frac{3}{-5}\) was derived from the previous step.
• The fraction \(\frac{3}{-5}\) simplifies to \(-\frac{3}{5}\) by moving the negative sign to a standard location, often in front of the fraction.

Simplification helps maintain uniformity and clarity in presenting solutions. It ensures that the solutions are ready for practical application or further use in more complex problem-solving.