Problem 64 - Elementary Algebra

Question

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables. Solve \(\star\) \(=\frac{\square+9 j}{\Delta}\) for \(j\).

Step-by-Step Solution

Answer

Question: Solve for the variable \(j\) in the following equation: \(=\frac{\square+9 j}{\Delta}\). Answer: \(j = \frac{\Delta \star - \square}{9}\)

Step 1: Isolate the term with \(j\)

Multiply both sides of the equation by \(\Delta\), this will move it to the left side and isolate the term with \(j\) on the right side. $$\Delta \star = (\square+9 j)$$

Step 2: Isolate \(j\)

Now, separate the \(9j\) term from \(\square\). To do this, subtract \(\square\) from both sides of the equation. $$\Delta \star - \square = 9j$$

Step 3: Solve for \(j\)

We are almost done. To solve for \(j\), divide both sides of the equation by 9. $$\frac{\Delta \star - \square}{9} = j$$ With this final step, we have found the solution: $$j = \frac{\Delta \star - \square}{9}$$

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and reorganizing algebraic expressions and equations to make them easier to work with or to solve them. This includes operations like adding, subtracting, multiplying, and dividing both sides of an equation by the same number or expression, as well as using properties like the distributive property to combine like terms.

Importance of Understanding Algebraic Manipulation

Understanding algebraic manipulation is crucial because it lays the foundation for solving various types of equations in algebra. It allows students to transition from simple numerical operations to more complex algebraic problem-solving. By mastering algebraic manipulation, students can break down seemingly complicated equations into more manageable parts. This skill not only aids in equation solving but also in simplifying expressions, factoring polynomials, and manipulating formulas in physical sciences and engineering.

Algebraic manipulation in action involves identifying the terms that include the variable we're trying to isolate and using the appropriate operations to move them to one side and the constant terms to the other. The goal is to simplify the equation to a point where the variable is alone on one side, making the equation easier to solve.

Isolate Variable

To isolate a variable means to manipulate an equation in such a way that the variable is by itself on one side of the equation, and everything else is on the other side. The process of isolating a variable is a central part of solving literal equations, which are equations where variables represent known values. It may require a series of algebraic manipulations depending on the complexity of the equation.

Steps for Isolating Variables

• Identify the target variable you need to isolate.
• Use inverse operations to undo any addition or subtraction around the variable.
• Apply inverse operations to undo any multiplication or division associated with the variable.
• Rearrange the equation if necessary, to have the variable alone on one side.

Isolating the variable is an important skill, not only for solving equations but also for rearranging formulas to solve for one variable in terms of others. This is particularly helpful in cases where you want to find the value of a variable given values for all the other variables in the formula.

Equation Solving Step by Step

Equation solving step by step refers to the systematic approach taken to find the value of an unknown variable in an equation. By breaking down the process into a sequence of smaller, more manageable steps, problems become more accessible and less intimidating. This methodical approach helps in detecting errors along the way and ensures that all the necessary algebraic manipulations are correctly applied.

Solving Equations in Phases

First, assess the equation and plan your strategy based on the operations and terms involved. Next, perform any simplifications necessary, such as combining like terms or removing parentheses. From there, use algebraic operations to isolate the variable step by step, remembering to perform the same operation to both sides of the equation to maintain balance. Finally, once the variable is isolated, verify your answer by plugging it back into the original equation, ensuring that the left side equals the right side.

Solving equations step by step is indispensable for students in mathematics courses, as it builds the problem-solving skills that will be used in more advanced topics and real-life applications. This methodical approach ensures that each step is logical and moves closer to the goal of finding the variable's value, thereby reducing the likelihood of errors in the process.