Problem 69
Question
Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve the equation \(\frac{6}{x+3}=\frac{4}{x-3}\) by multiplying both sides by the LCD.
Step-by-Step Solution
Verified Answer
Yes, the statement 'I can solve the equation \(\frac{6}{x+3}=\frac{4}{x-3}\) by multiplying both sides by the LCD' makes sense, as demonstrated in the step-by-step solution.
1Step 1: Analysis of the statement
We first analyze the given statement. Rational equations can be solved by using the LCD to eliminate the denominators. Therefore, the statement 'I can solve the equation \(\frac{6}{x+3}=\frac{4}{x-3}\) by multiplying both sides by the LCD' initially appears to make sense.
2Step 2: Determining the LCD
Determine the LCD for \(x+3\) and \(x-3\). The LCD of these two expressions is their product, \((x+3)(x-3)\). So, yes, it is entirely feasible to solve the equation by multiplying both sides by the LCD.
3Step 3: Implementing the LCD
To confirm whether the statement makes sense, let's multiply both sides of the equation by the LCD, \((x+3)(x-3)\). We get \((x+3)(x-3)*\frac{6}{x+3} = (x+3)(x-3)*\frac{4}{x-3}\). Simplifying, we get \(6(x-3)=4(x+3)\).
4Step 4: Solving the simplified equation
We can now solve this equation normally. By doing so, we can verify that this method indeed provides a valid solution to the original equation, hence the statement does make sense.
Key Concepts
Least Common DenominatorLCD in AlgebraEquation Solving StrategiesRational Expression Multiplication
Least Common Denominator
When working with rational equations, the least common denominator (LCD) is the smallest expression that all denominators can divide into without leaving a remainder. Its primary use is to combine fractions or rational expressions that have different denominators.
To find the LCD, we consider the factors of each denominator and select each factor at its highest power that appears in any of the denominators. For instance, if you have denominators like \(x+3\) and \(x-3\), the LCD would not simply be \(x\), since it does not account for the distinct plus and minus expressions. Instead, the LCD combines the factors to encompass both, resulting in \(x+3)(x-3)\).
Notably, when applying the LCD to clear denominators, it's essential to ensure the LCD isn't zero since division by zero is undefined. This is why, before solving equations, we must define the domain in which the variable \(x\) can operate.
To find the LCD, we consider the factors of each denominator and select each factor at its highest power that appears in any of the denominators. For instance, if you have denominators like \(x+3\) and \(x-3\), the LCD would not simply be \(x\), since it does not account for the distinct plus and minus expressions. Instead, the LCD combines the factors to encompass both, resulting in \(x+3)(x-3)\).
Notably, when applying the LCD to clear denominators, it's essential to ensure the LCD isn't zero since division by zero is undefined. This is why, before solving equations, we must define the domain in which the variable \(x\) can operate.
LCD in Algebra
In algebra, the LCD plays a critical role in simplifying and solving equations involving fractions or rational expressions. When you encounter an equation like \(\frac{6}{x+3}=\frac{4}{x-3}\), you can use the LCD to eliminate the denominators, turning the rational equation into a simpler algebraic equation.
By multiplying every term by the LCD, you are essentially synchronizing the denominators, which allows them to cancel out. In our example, since the LCD is \(x+3)(x-3)\), multiplying each side by this expression clears the fractions and simplifies the problem. After multiplication and simplification, variables and constants can be moved and combined to solve for \(x\) with more familiar algebraic methods.
By multiplying every term by the LCD, you are essentially synchronizing the denominators, which allows them to cancel out. In our example, since the LCD is \(x+3)(x-3)\), multiplying each side by this expression clears the fractions and simplifies the problem. After multiplication and simplification, variables and constants can be moved and combined to solve for \(x\) with more familiar algebraic methods.
Equation Solving Strategies
Solving equations effectively requires a toolbox of equation solving strategies. One such tool is the use of the LCD to simplify rational equations. The steps to this strategy often involve identifying the LCD, multiplying each term of the equation by the LCD to eliminate fractions, and then solving the resulting equation using strategies such as isolating the variable, combining like terms, and employing the distributive property.
Additional Tips:
- Always check your solution by plugging it back into the original equation to ensure it does not make the denominator zero.
- Remember to simplify expressions whenever possible to make the equation easier to work with.
- Consider all possible solutions, including negatives, fractions, and irrational numbers when appropriate.
Rational Expression Multiplication
Multiplying rational expressions involves a similar process to multiplying numerical fractions. Where you multiply the numerators together and the denominators together. But before jumping into the multiplication of rational expressions, it's wise to factor both numerators and denominators first.
This preliminary step may reveal common factors in the numerator and denominator, which can be cancelled out to simplify the expression further. When you multiply each term in a rational equation by the LCD, as in the provided exercise, you make use of this multiplication principle. Once the denominators have been cleared, what remains is a simpler expression or an algebraic equation that is often much easier to solve.
This preliminary step may reveal common factors in the numerator and denominator, which can be cancelled out to simplify the expression further. When you multiply each term in a rational equation by the LCD, as in the provided exercise, you make use of this multiplication principle. Once the denominators have been cleared, what remains is a simpler expression or an algebraic equation that is often much easier to solve.
Example:
- Let's consider the multiplication \(\frac{a}{b})\times(\frac{c}{d})\), you would end up with \(\frac{ac}{bd}\).
- In the context of our exercise, multiplying \(\frac{6}{x+3}\) by \(x-3\) and \(\frac{4}{x-3}\) by \(x+3\) after factoring and simplifying, you manage to clear the denominators, resulting in an equation \(6(x-3)=4(x+3)\) that is ready to solve.
Other exercises in this chapter
Problem 69
Add or subtract as indicated. Simplify the result, if possible. $$\frac{9 x+3}{x^{2}-x-6}+\frac{x}{3-x}$$
View solution Problem 69
Perform the indicated operation or operations. $$\frac{x^{2}+x z+x y+y z}{x-y} \div \frac{x+z}{x+y}$$
View solution Problem 70
perform the indicated operation or operations. Simplify the result, if possible. $$\frac{y}{a x+b x-a y-b y}-\frac{x}{a x+b x-a y-b y}$$
View solution Problem 70
Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-4}{2-x}$$
View solution