Problem 59 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{4}{m}-\frac{5}{m-3}=7 $$

Step-by-Step Solution

Answer

Question: Solve the given rational equation for m: $$\frac{4}{m}-\frac{5}{m-3} = 7$$ Answer: After solving the given rational equation, the value of m is $$m = \frac{4}{7}$$.

Step 1: Identify the common denominator

Recognize that the least common denominator (LCD) for this equation is m(m-3) as it includes the denominators of both fractions in the equation.

Step 2: Eliminate the denominators

Multiply both sides of the equation by the LCD (m(m-3)) to eliminate the denominators. $$ m(m-3)\bigg(\frac{4}{m}-\frac{5}{m-3}\bigg) = 7(m(m-3)) $$

Step 3: Distribute and simplify

Distribute the LCD into the equation and simplify. $$ (m-3)(4)-(m)(5) = 7(m^2-3m) $$ $$ 4m-12-5m = 7m^2-21m $$

Step 4: Rearrange the equation

Rearrange the equation to form a quadratic equation in the form of \(ax^2+bx+c=0\) $$ 7m^2 -(16m) +12 = 0 $$

Step 5: Solve for m

Use whatever method you prefer to solve the quadratic equation (factoring, quadratic formula, etc). In this case, it can easily be factored: $$ (7m - 4)(m - 3) = 0 $$ Therefore, the possible values for m are: $$ m = \frac{4}{7}, 3 $$ However, since the denominator of the second term in the original equation was (m-3), m cannot be 3 as it would make the denominator equal to zero, which is not allowed in rational equations. Hence, the only solution is: $$ m = \frac{4}{7} $$

Key Concepts

Least Common Denominator

Understanding the least common denominator (LCD) is vital when solving rational equations that involve fractions. A rational equation is an equation containing at least one fraction whose numerator and denominator are polynomials. The LCD is the smallest common multiple of these denominators and is used to combine the fractions on a common footing without losing equality.

For instance, in the problem \(\frac{4}{m}-\frac{5}{m-3}=7\), identifying \(m(m-3)\) as the LCD is crucial because it allows you to multiply each term of the equation by this number to eliminate the fractions. This step ensures you're working with a polynomial equation which is easier to solve, without changing the solutions to the original equation. Mastering how to find the LCD can simplify complex rational equations and is a skill that will be a foundation for much of algebra.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). These equations are fundamental in algebra and encompass a broad range of mathematical areas. To solve a quadratic equation, one must find the value of \(x\) that makes the equation true.

There are several standard methods for solving quadratic equations, such as factoring, completing the square, and using the quadratic formula. In the problem provided, rearranging the simplified equation into a quadratic format, \(7m^2 -(16m) +12 = 0\), is a step towards finding the possible values of \(m\). Once in this form, you can proceed with the most suitable method for solving the quadratic equation.

Factoring Quadratic Expressions

Factoring is a process that involves rewriting a quadratic expression as the product of simpler expressions. When you factor a quadratic equation like \(7m^2 - 16m +12 = 0\), you look for two binomials that, when multiplied together, will give you the original quadratic expression. This concept is essential not just for solving equations but for understanding algebraic expressions more deeply.

Factoring becomes useful when the quadratic expression can be easily decomposed into a product of binomials, as we see with the problem at hand: \(7m^2 - 16m + 12 = 0\) becomes \( (7m - 4)(m - 3) = 0\). Once factored, the Zero Product Property indicates that if a product equals zero, at least one of the factors must be zero. This realization leads to viable solutions for \(m\), which should then be checked against the original equation to confirm they don't result in undefined expressions (like a zero denominator).