Problem 18
Question
Solve \(\frac{12}{m-4}+5=\frac{3 m}{m-4}\).
Step-by-Step Solution
Verified Answer
Question: Determine the solution, if any, for the equation \(\frac{12}{m-4}+5=\frac{3 m}{m-4}\).
Answer: There is no solution for this equation.
1Step 1: Eliminate Denominators
To eliminate the denominators, first find the least common denominator (LCD). Here, it is \((m-4)\). Now, multiply both sides of the equation by the LCD to get rid of the fractions:
\((m-4)[\frac{12}{m-4}+5]=(m-4)\frac{3m}{m-4}\).
2Step 2: Simplify the Equation
Distribute the \((m-4)\) on both sides of the equation:
\((m-4)\cdot\frac{12}{m-4}+(m-4)\cdot5=(m-4)\cdot\frac{3m}{m-4}\).
Since the denominators cancel out, you are then left with:
\(12+5(m-4)=3m\).
Now, distribute the \(5\) on the left side:
\(12+5m-20=3m\).
Combining the constant terms on the left side, we get:
\(5m-8=3m\).
3Step 3: Solve for m
To solve for \(m\), first, we will move \(3m\) to the left side in order to have all \(m\) terms on one side of the equation:
\(5m-3m=8\).
Simplifying, we get:
\(2m=8\).
Now, divide both sides by \(2\) to isolate \(m\):
\(m=\frac{8}{2}\).
Finally, we obtain the value of \(m\):
\(m=4\).
However, substituting this value back into the original equation, we see that we get a denominator of \(0\), which is not allowed. Therefore, we can conclude that there is no solution for this equation.
Key Concepts
Least Common DenominatorEquation SimplificationAlgebraic FractionsNo Solution Conditions
Least Common Denominator
Understanding the least common denominator (LCD) is critical when solving equations involving algebraic fractions. The LCD is the smallest expression that all denominators within an equation can divide into. In the context of solving rational equations, finding the LCD allows us to combine fractions by creating a common baseline.
This approach simplifies the equation by converting it to a form without fractions, easing further manipulation. For the given exercise, where we have \frac{12}{m-4}+5=\frac{3 m}{m-4}\], the LCD is simply \(m-4\). It's the shared denominator of the fractions, which allows us to multiply through to eliminate the fractions and move forward to solving the equation.
This approach simplifies the equation by converting it to a form without fractions, easing further manipulation. For the given exercise, where we have \frac{12}{m-4}+5=\frac{3 m}{m-4}\], the LCD is simply \(m-4\). It's the shared denominator of the fractions, which allows us to multiply through to eliminate the fractions and move forward to solving the equation.
Equation Simplification
Simplification of an equation is an essential step in solving mathematical problems. It involves reducing the equation to its simplest form to make it easier to solve. The goal is to condense the equation while maintaining its originality. For rational equations, this often involves clearing the fractions as seen in our example.
After multiplying by the least common denominator, we distribute terms to cancel out denominators and then combine like terms. In the given exercise, by distributing \(5\) and combining constant terms, we simplify the equation to \(5m - 8 = 3m\). This clearer form showcases the relationship between variables and constants, leading us directly towards solving for \(m\).
After multiplying by the least common denominator, we distribute terms to cancel out denominators and then combine like terms. In the given exercise, by distributing \(5\) and combining constant terms, we simplify the equation to \(5m - 8 = 3m\). This clearer form showcases the relationship between variables and constants, leading us directly towards solving for \(m\).
Algebraic Fractions
Algebraic fractions, also known as rational expressions, are fractions that contain one or more algebraic expressions in their numerator and/or denominator. Solving equations involving these fractions can seem daunting, but with the right techniques, it becomes manageable.
To handle such equations, we seek to isolate the variable of interest. This is done by finding a common denominator, which allows us to merge fractions and eventually rid the equation of any fractional parts. It's essential to remember that the denominators should not be zero, as division by zero is undefined within the realm of mathematics. Our example exercise illustrates this perfectly, as we work to combine \(\frac{12}{m-4}\) and \(\frac{3 m}{m-4}\) under a common denominator.
To handle such equations, we seek to isolate the variable of interest. This is done by finding a common denominator, which allows us to merge fractions and eventually rid the equation of any fractional parts. It's essential to remember that the denominators should not be zero, as division by zero is undefined within the realm of mathematics. Our example exercise illustrates this perfectly, as we work to combine \(\frac{12}{m-4}\) and \(\frac{3 m}{m-4}\) under a common denominator.
No Solution Conditions
In the realm of algebra, equations can sometimes have no solution. These conditions often arise when operations on an equation lead to a contradiction or an undefined scenario, such as dividing by zero. Such is the case in our current exercise.
Even though we found that \(m=4\), substituting this back into the original equation would result in division by zero \((m-4)\), which is impermissible. Therefore, we declare the equation unsolvable. This understanding is critical as it reinforces the concept that not all equations will have a numerical solution, and recognizing ‘no solution’ scenarios demonstrates a deeper comprehension of algebraic principles.
Even though we found that \(m=4\), substituting this back into the original equation would result in division by zero \((m-4)\), which is impermissible. Therefore, we declare the equation unsolvable. This understanding is critical as it reinforces the concept that not all equations will have a numerical solution, and recognizing ‘no solution’ scenarios demonstrates a deeper comprehension of algebraic principles.
Other exercises in this chapter
Problem 17
Add or Subtract the following rational expressions. $$ \frac{x}{x+4}-\frac{x-2}{3 x-3} $$
View solution Problem 17
For the following problems, reduce each rational expression to lowest terms. $$ \frac{20 a^{4} b^{4}}{4 a b^{2}} $$
View solution Problem 18
For the following problems, perform the divisions. $$ \frac{3 x^{2}-6 x}{-3} $$
View solution Problem 18
One pipe can drain a pool in 12 hours. Another pipe can drain the pool in 15 hours. How long does it take both pipes working together to drain the pool?
View solution