Problem 16 - Elementary Algebra

Question

Solve each of the equations. $$\frac{-9}{x+1}=\frac{-8}{x+5}$$

Step-by-Step Solution

Answer

The solution is \(x = -37\).

Step 1: Identify the Cross Multiplication Method

To solve the equation \( \frac{-9}{x+1} = \frac{-8}{x+5} \), we'll use the method of cross-multiplication. This involves multiplying across the equation diagonally.

Step 2: Apply Cross Multiplication

Multiply both sides diagonally to eliminate the fractions. This gives: \(-9(x + 5) = -8(x + 1)\).

Step 3: Distribute the Constants

Expand the expressions on both sides: \(-9(x + 5) = -9x - 45\) and \(-8(x + 1) = -8x - 8\). Thus, the equation becomes \(-9x - 45 = -8x - 8\).

Step 4: Rearrange the Equation

Move all terms involving \(x\) to one side and constants to the other side by adding \(9x\) to both sides: \(-45 = x - 8\).

Step 5: Solve for x

Add \(8\) to both sides to isolate \(x\): \(-45 + 8 = x\). Simplifying gives \(-37 = x\).

Key Concepts

Solving Rational Equations

Rational equations are mathematical expressions that involve fractions and variables. These equations can often seem challenging, but by using effective strategies such as cross multiplication, they become easier to solve.
In rational equations like \( \frac{-9}{x+1} = \frac{-8}{x+5} \), we aim to eliminate fractions to simplify the equation. Cross multiplication is a key method for this. It allows us to find a common basis for comparison, helping us transition from a fractional equation to a solvable linear form.
The essential step in cross multiplying involves multiplying both numerators and denominators across the equality sign, diagonally. For instance, in this equation, we multiply \(-9\) by \( (x+5) \) and \(-8\) by \( (x+1) \). By doing so, the equation is transformed into \(-9(x + 5) = -8(x + 1)\).
This transformation plays a crucial role in simplifying the equation. It removes the fractions, allowing us to focus on solving a simpler linear equation, devoid of any fractions.

Step by Step Algebra Solutions

Understanding the step by step algebraic solutions is vital in mastering equations and improving problem-solving skills.
Here, we start with the equation \(\frac{-9}{x+1} = \frac{-8}{x+5}\), moving forward through cross multiplication to convert into \(-9(x + 5) = -8(x + 1)\).

• First, distribute the constants by multiplying them with each term within the parentheses. This results in \(-9x - 45\) on the left and \(-8x - 8\) on the right.
• Next, collect like terms by moving all terms containing \(x\) to one side and numerical constants to the other. This involves adding \(9x\) to both sides of the equation, which simplifies our problem significantly.

Once you have isolated the variable terms, it's a matter of simplifying to find \(x = -37\). Each step builds on the previous, offering a structured approach that facilitates understanding and accuracy in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and subtraction). In the context of solving equations, they represent a quantity that can change value depending on the input of variables.
Expressions such as \( -9(x + 5) \) and \( -8(x + 1) \) are algebraic because they involve variables \(x\). These can be simplified or transformed to aid in solving equations. Simplifying involves expanding the expression: multiply constants across terms within parentheses to separate values.

• For our equation, \(-9(x + 5) \) becomes \(-9x - 45\), and \(-8(x + 1) \) becomes \(-8x - 8\).

By simplifying these expressions, we change from a complex equation to a simpler linear form that can be easily solved by traditional algebraic methods. This process is crucial, as it supports the solution of more complex equations, laying a foundation for understanding bigger mathematical concepts.