Problem 9 - Beginning and Intermediate Algebra

Question

Solve each equation. Check the solutions. \(-\frac{3}{x}-\frac{28}{x^{2}}=0\)

Step-by-Step Solution

Answer

x = -9.33 (approximately)

Step 1 - Identify the common denominator

The common denominator of the terms \(-\frac{3}{x}\) and \(-\frac{28}{x^{2}}\) is \(x^2\).

Step 2 - Multiply through by the common denominator

Multiply every term in the equation by \(x^2\) to eliminate the denominators: \[x^2 \cdot \left( -\frac{3}{x} \right) + x^2 \cdot \left( -\frac{28}{x^2} \right) = 0 \]

Step 3 - Simplify the equation

Simplify each term: \[-3x - 28 = 0 \]

Step 4 - Solve for \(x\)

Add 28 to both sides of the equation: \[-3x = 28 \]

Step 5 - Isolate \(x\)

Divide both sides by \(-3\): \[x = -\frac{28}{3} = -9.33 (approximately) \]

Step 6 - Verify the solution

Substitute \(x = -\frac{28}{3}\) back into the original equation to check if the left side equals zero: -\frac{3}{-\frac{28}{3}}-\frac{28}{{\left(-\frac{28}{3}\right)}^{2}} = 0.

Key Concepts

Common Denominator

A common denominator is a common multiple of the denominators of two or more fractions. Having a common denominator helps simplify equations by eliminating the denominators. To find the common denominator, identify the least common multiple (LCM) of all the denominators involved. In our exercise, the terms \(-\frac{3}{x}\) and \(-\frac{28}{x^{2}}\) have denominators of \(x\) and \(x^2\), respectively. The least common multiple of these denominators is \(x^2\). This common denominator allows us to later eliminate the denominators entirely, making the equation easier to handle.

Eliminating Denominators

Eliminating denominators involves multiplying each term in the equation by the common denominator. This process helps convert the rational equation into a simpler polynomial equation. In our example, after determining that \(x^2\) is the common denominator, we multiply each term in the equation \(-\frac{3}{x}-\frac{28}{x^{2}}=0\) by \({x^2}\). This gives us: \[x^2 \cdot \left(-\frac{3}{x}\right) + x^2 \cdot \left(-\frac{28}{x^2}\right) = 0\] When simplified, it becomes: \(-3x - 28 = 0\). This transformed equation no longer has any denominators, easing the solving process.

Isolating Variables

Isolating the variable in an equation involves solving for that variable by performing operations that simplify the equation step-by-step. After simplifying to \(-3x - 28 = 0\), the goal is to isolate \(x\). Start by moving all constants to the other side of the equation. In this case, add 28 to both sides: \(-3x = 28\). Next, to solve for \(x\), divide each side of the equation by \(-3\): \[x = -\frac{28}{3} \] This isolates \(x\) and shows the solution to the equation. The approximate value of \(x\) is \(-9.33\).

Substitution Check

A substitution check verifies the solution by substituting the value of the variable back into the original equation. This step ensures the solution is correct. For our example, substitute \(x = -\frac{28}{3}\) back into the original equation \(-\frac{3}{x} - \frac{28}{x^2}=0\): \[-\frac{3}{-\frac{28}{3}} - \frac{28}{{\left(-\frac{28}{3}\right)}^2} = 0\] Simplifying inside the fractions: \[-\frac{3}{-9.33} - \frac{28}{\left( -9.33 \right)^{2}} \ = \underline{\phantom{xxx}} -\frac{3 \times 3}{-28} - \frac{28}{\frac{784}{9}} \ = \underline{\phantom{xxx}} \frac{9}{28} - \frac{28 \cdot 9 }{784} \ = 0\] Hence, the solution \(x = -\frac{28}{3}\) is verified.