Problem 21 - Technical Mathematics with Calculus

Question

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(\frac{x-1}{8}-\frac{x+1}{18}=1\)

Step-by-Step Solution

Answer

After following the steps to combine fractions, simplify the equation, and solve for x, the final answer is that x equals 17.

Step 1: Find the Least Common Denominator (LCD)

To combine the fractions, we need a common denominator. The least common denominator (LCD) for 8 and 18 is their least common multiple (LCM). To find the LCM of 8 and 18, we list their multiples and determine that their LCM is 72. Thus, we will convert each fraction to have a denominator of 72.

Step 2: Rewrite each fraction with the LCD

Rewrite the fractions \(\frac{x-1}{8}\) and \(\frac{x+1}{18}\) with the LCD as the new denominator. For the first fraction, multiply the numerator and denominator by 9 to get \(\frac{9(x-1)}{72}\). For the second fraction, multiply the numerator and denominator by 4 to get \(\frac{4(x+1)}{72}\).

Step 3: Rewrite the equation with the new fractions and simplify

Replace the original fractions with the new fractions to get \(\frac{9(x-1)}{72} - \frac{4(x+1)}{72} = 1\). Because the denominators are the same, combine the numerators and keep the common denominator as 72.

Step 4: Combine like terms and solve for x

Combine the numerators to get \(\frac{9x - 9 - 4x - 4}{72} = 1\). This simplifies to \(\frac{5x - 13}{72} = 1\). Cross multiply to remove the fraction: \(5x - 13 = 72\).

Step 5: Isolate x

To isolate x, we need to add 13 to both sides of the equation: \(5x - 13 + 13 = 72 + 13\), which simplifies to \(5x = 85\). Then, divide both sides by 5 to solve for x: \(x = \frac{85}{5}\).

Step 6: Simplify the fraction for x

Simplifying the fraction \(\frac{85}{5}\) we get \(x = 17\). Although the instruction is to leave the answer in fractional form, in this case, the fraction simplifies to a whole number.

Key Concepts

Least Common Denominator

When faced with multiple fractions in an equation, finding the Least Common Denominator (LCD) is a crucial step in solving the problem. The LCD is the smallest number that each denominator can divide into without a remainder. It allows us to combine fractions in a way that makes the rest of the arithmetic straightforward.

Finding the LCD often involves listing the multiples of each denominator until you find the smallest common one. Let's consider the denominators 8 and 18 from our exercise. The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, and so on, while those of 18 are 18, 36, 54, 72, and so forth. The first common multiple we encounter is 72, making it our LCD. Using this approach simplifies the equation and ultimately reveals one fraction with a common denominator that we can solve.

Simplifying Fractions

To simplify fractions means to reduce them to their most basic form where the numerator and denominator are as small as possible. When we say 'as small as possible', we're looking to find the largest common factor that both the numerator and the denominator share and divide them by it.

For example, in our solution step where we need to simplify the fraction \(\frac{85}{5}\), we observe that both 85 and 5 share a common factor – which is 5. Dividing top and bottom by this common factor gives us \(\frac{85\div5}{5\div5}\), simplifying down to \(\frac{17}{1}\), or simply 17. This process is not only crucial in obtaining the final solution but also in making intermediate steps easier to manage.

Linear Equation Step by Step Solution

Solving linear equations step by step involves a systematic approach to manipulate the equation into a form where the variable is isolated on one side, giving the solution. As illustrated in the original exercise, this can involve several stages, each with its own purpose, leading to the final answer.

The journey typically starts by addressing any fractions through finding a common denominator, followed by rewriting the equation with combined terms. The next step involves combining like terms - in other words, grouping and simplifying the x's and the constants separately. Once the equation consists of a single fraction equal to a number, as in our example \(\frac{5x - 13}{72} = 1\), we employ cross-multiplication to eliminate the fractions entirely. Finally, we isolate the variable (x in this case) by performing arithmetic operations such as addition, subtraction, multiplication, or division until x stands alone, representing the solution to the equation.