Problem 42 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solutions. \(\frac{x}{9}-\frac{8}{x}=\frac{1}{9}\)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x}{9}-\frac{8}{x}=\frac{1}{9}\) is \(x=4\).

Step 1: Clear out the fractions

To clear out the fractions, start by multiplying each term by the common denominator 9x. This gives \(x*9x\)-\(8*9\)=\(1*9x\), which simplifies to \(9x^2-72=9x\).

Step 2: Rearrange the equation

Move all the terms to one side of the equation to set it to zero. This gives \(9x^2 - 9x - 72 = 0\).

Step 3: Factor the quadratic equation

Factor the equation as \((3x-12)(3x+6)=0\).

Step 4: Set each factor to zero to solve for x

Setting each factor equal to zero gives \(3x-12=0\) and \(3x+6=0\). Solving both equations results in \(x=4\) from the first equation and \(x=-2\) from the second equation. But since x was originally in the denominator, \(x ≠ -2\) as it would make the original equation undefined. Therefore, we discard \(x = -2\).

Step 5: Check the solution

We substitute the solution back into the original equation. Upon substitution, the left-hand side becomes \(\frac{4}{9}-\frac{8}{4}\) which simplifies to \(\frac{1}{9}\), equal to right-hand side.

Key Concepts

Clearing Fractions in Equations

When faced with fractions in equations, as in \( \frac{x}{9}-\frac{8}{x}=\frac{1}{9} \), the first step is to eliminate fractions by finding a common denominator and multiplying each term. This process is known as 'clearing fractions'.

In our example, the common denominator for \( \frac{x}{9} \) and \( \frac{8}{x} \) is 9x. By multiplying each term by 9x, we effectively clear the fractions because we are aiming for an equation with integer coefficients. This simplifies our equation to \(9x^2-72=9x\).

It's pivotal to remember to multiply every term—including constants and variables—by the common denominator to keep the equation balanced. After multiplying, ensure you simplify, if necessary, to keep the equation neat, setting the stage for the next step: factoring the quadratic equation.

Factoring Quadratic Equations

Factoring is a method used to solve quadratic equations, such as \(9x^2 - 9x - 72 = 0\), by expressing them as a product of simpler binomials. In this case, the quadratic trinomial is broken down into \(3x-12\) and \(3x+6\).

Factoring requires identifying a pair of numbers that, when multiplied, give the product of the a-term and c-term (from ax^2 + bx + c), and when added, give the b-term.

Identifying the Trial Numbers

Here, we look for two numbers that multiply to -216 (\(-72\times 3\)) and add to -9 (the coefficient of x). Those numbers are -18 and +6. We then rewrite the middle term of the quadratic equation using these two numbers, followed by grouping and factoring out the common factor from each binomial.

This skill is essential in simplifying equations to a form that is much easier to solve, since after factoring, we often just need to set each factor equal to zero and solve for x, greatly simplifying the original problem.

Quadratic Equation Solutions

Once we have factored the quadratic equation, as in our example to \(3x-12\) and \(3x+6\), we can find the solutions by setting each factor equal to zero. This gives us potential solutions for x.

In our exercise, setting \(3x-12=0\) leads to \(x=4\), and \(3x+6=0\) leads to \(x=-2\). However, it's crucial to check these solutions in the original equation to ensure they do not render any denominators zero, which would make them undefined.

Importance of Validation

In this case, \(x=-2\) is not a valid solution because it makes the original denominator zero. Hence, we only accept \(x=4\) as the solution. Always plug the solutions back into the original equation to validate them. This check confirms the correctness of our solutions and that they work within the constraints of the original problem.