Problem 45 - Algebra for College Students

Question

Solve each equation. $$\frac{x}{9}=\frac{-20}{9 x}+1$$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x =5\).

Step 1: Eliminate the Fraction

Multiply both sides of the equation by 9 to eliminate the fraction. This gives: \[ 9 \times \frac{x}{9} = 9 \times \frac{-20}{9x} + 9 \times 1 \]

Step 2: Simplify the Equation

After multiplying, simplify the left and right sides: \[ x = \frac{-20}{x} + 9 \]

Step 3: Eliminate the Fraction Again

Multiply both sides of the equation by \(x\) to eliminate the fraction: \[ x \times x = x \times \frac{-20}{x} + x \times 9 \]. This simplifies to: \[ x^2 = -20 + 9x \]

Step 4: Rearrange into a Standard Form Quadratic Equation

Move all terms to one side of the equation to set it equal to zero: \[ x^2 - 9x + 20 = 0 \]

Step 5: Factor the Quadratic Equation

Factor the quadratic equation: \[ (x - 4)(x - 5) = 0 \]

Step 6: Solve for x

Set each factor equal to zero and solve for \(x\): \[ x - 4 = 0 \] and \[ x - 5 = 0 \]. Therefore, \[ x = 4 \] and \[ x = 5 \].

Key Concepts

Quadratic Equation

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\). Quadratic equations appear frequently in algebra and other areas of mathematics. They include a variable raised to the second power \(x^2\). The standard form usually starts with the highest degree term (\(x^2\)). Quadratic equations are used to describe parabolas, optimize functions, and solve various real-world problems. You will often need to use methods like factoring, completing the square, or applying the quadratic formula to solve these equations.

Factoring

Factoring is the process of breaking down an equation into simpler expressions that can be multiplied together to get back to the original equation. In the exercise, after rearranging the equation into standard form \(x^2 - 9x + 20 = 0\), factoring allows us to express it as \((x-4)(x-5)=0\). Factoring can simplify solving by turning the quadratic into a product of linear expressions we can easily solve. These simpler expressions give us the roots of the quadratic equation when set equal to zero.

Fraction Elimination

Eliminating fractions is a crucial step when solving equations involving fractions, as it simplifies the equation. In the given problem \(\frac{x}{9}=\frac{-20}{9x}+1\), we eliminate fractions by multiplying everything by 9, resulting in\( x = \frac{-20}{x} + 9 \). This process makes the equation easier to handle. Later, we again eliminate fractions by multiplying both sides by \(x\), which helps to transform the equation into a quadratic form. These steps are crucial for simplifying and solving equations involving fractions.

Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). It allows for uniform methods of solving, such as factoring, the quadratic formula, or completing the square. In our exercise, we reached \(x^2 - 9x + 20 = 0\), which is in standard form. This arrangement makes it easy to apply the quadratic solving techniques mentioned earlier. Rewriting the equation in this standardized way is a key step in working with quadratic equations efficiently.