Problem 70 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -3 x=-1 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{1}{3}\).

Step 1: Understand the Equation

The given equation is \(-3x = -1\). Our goal is to solve for \(x\), which means we need to find the value of \(x\) such that the left side of the equation equals the right side.

Step 2: Isolate the Variable

To isolate \(x\), we need to divide both sides of the equation by \(-3\) which is the coefficient of \(x\). This can be represented as follows: \[x = \frac{-1}{-3}\]

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{-1}{-3}\). Since dividing two negative numbers results in a positive number, the simplified form of the fraction is \(\frac{1}{3}\).

Step 4: Verify the Solution

Plug the solution back into the original equation to verify it. Substitute \(x = \frac{1}{3}\) into \(-3x = -1\): \(-3 \times \frac{1}{3} = -1\). Indeed, both sides equal \(-1\), confirming that the solution is correct.

Key Concepts

Equation Isolation

In solving linear equations, the primary goal is to isolate the variable we are solving for, in this case, the variable \(x\). Isolation means getting \(x\) by itself on one side of the equation. In our example, the original equation is \(-3x = -1\). Here, \(x\) is multiplied by \(-3\).

To isolate \(x\), perform the opposite operation on both sides. Since multiplication is involved, we will divide both sides by \(-3\). This results in:

• The equation transforms to \(x = \frac{-1}{-3}\).
• Both sides are treated equally to maintain the balance of the equation.

By following this process, \(x\) stands alone on one side, ready for the next steps of solving.

Fraction Simplification

Simplifying fractions is a crucial step to keep your solutions neat and clear. When we arrived at \(x = \frac{-1}{-3}\), we need to simplify the fraction. Simplification involves making a fraction as simple as possible while keeping its value unchanged.

Here, both the numerator and denominator are negative. Recall the rule: dividing two negative numbers results in a positive number. Therefore, \(\frac{-1}{-3}\) simplifies to \(\frac{1}{3}\).

• This step ensures the solution is easy to read and understand.
• It retains the value while making the mathematical expression more straightforward.

Fractions simplified correctly lead to fewer errors in subsequent calculations.

Verification of Solution

Verification is a vital final step in solving equations. It assures that the solution works within the context of the original equation. For our resolved solution, \(x = \frac{1}{3}\), verification involves substituting \(x\) back into the initial equation, \(-3x = -1\).

When you substitute \(x = \frac{1}{3}\), you perform the multiplication:

• Calculate \(-3 \times \frac{1}{3}\) which yields \(-1\).
• The output matches the right side of the original equation.

Since both sides match accurately, this confirms our solution is correct.

Verification reinforces the accuracy and confidence in your result.