Problem 43 - Algebra 1

Question

Describe the errors. $$ \begin{aligned} 2(4 b-3) &=8 b-6 \\ 8 b-6 &=8 b-6 \\ -6 &=-6 \\ b &=-6 \end{aligned} $$

Step-by-Step Solution

Answer

The error lies in the second line of the calculation where \(8b - 6 = 8b - 6\) is equated to \(-6 = -6\) without isolating 'b'. The equation is valid for all values of 'b' and hence there are infinite solutions, not \(b = -6\).

Step 1: Check Calculation

Begin by looking at the first line of the equation, \(2(4b - 3) = 8b - 6\). The left-hand side (LHS) simplifies correctly to the right-hand side (RHS). So, no error at this point.

Step 2: Identify The Errors

On the second line where \(8b - 6 = 8b - 6\) was directly equated to \(-6 = -6\), the correct procedure was not followed. Normally, terms with the 'b' variable on both sides of the equation would be collected on one side to isolate the variable, 'b'.

Step 3: Demonstrate the Correct Procedure

Let's follow the correct procedure from step2. Subtract \(8b\) from both sides to isolate the variable 'b', giving the equation \(-6 = -6\). However, given that there is no 'b' term remaining, we conclude that this equation condition is valid for all values of 'b', and there are infinite solutions for 'b'.

Key Concepts

Equation Solving Errors

Identifying and understanding errors when solving algebraic equations is crucial for mastering algebra. A common mistake is the improper isolation of variables.
In our exercise, an error occurs when the equation is simplified from \(8b - 6 = 8b - 6\) directly to \( -6 = -6\), without properly dealing with the terms containing the variable \(b\). The student incorrectly jumps to the conclusion that \(b = -6\), which does not follow from the previous step. Instead, when both sides of the equation contain the same terms, they should cancel each other out.
It is important to understand that an equation like \(8b - 6 = 8b - 6\) indicates that all values of \(b\) are solutions, because the variable terms cancel each other out, and we are left with an identity (a true statement for all values of the variable). This oversight is a good reminder to carefully handle each step of simplification.
To avoid such errors:

• Ensure you perform the same operations on both sides of the equation.
• Combine like terms before attempting to isolate the variable.
• Check each step for accuracy before moving on to the next.

Linear Equations

The essence of solving linear equations is finding the value(s) of the variable(s) that make the equation true. A linear equation is an equation where the variable—such as \(b\) in our example—is raised to the first power and thus forms a straight line when graphed on a coordinate plane.

Characteristics of Linear Equations

Linear equations generally take the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. The equation \(2(4b - 3) = 8b - 6\) from the exercise simplifies to this standard form.

Methods for Solving Linear Equations

• Cross multiplication: Used when the equation is in fraction form.
• Substitution: Substitute one equation into another when working with systems of equations.
• Elimination: Add or subtract equations in a system to eliminate one variable.
• Graphical method: Plot each equation on a graph and identify the intersection point.

When solving a linear equation, you're often finding the intersection of two lines, which correlates to the solution(s) that satisfy both equations. In our exercise, instead of being a single point, every point on the line is a solution, reflecting the infinite solutions for the value of \(b\).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the context of linear equations, these expressions can be simplified and manipulated to solve for the unknown variable(s).
In our example, \(2(4b - 3)\) is an algebraic expression that simplifies to \(8b - 6\). This process is an important step in solving the equation. Understanding how to expand, factor, and simplify these expressions is essential to algebra.

Key Components of Algebraic Expressions

• Variables: Symbols that represent unknown values which we often need to solve for.
• Coefficients: Numerical or constant factors of the variables in an expression.
• Constants: Numbers that stand alone without any variables.
• Operators: The plus (+), minus (-), multiplication (*), and division (/) symbols that tell us what to do with the components of the expression.

The ability to rearrange and simplify these expressions is vital for solving equations effectively. Not only will it help in finding the value of a variable, but it also comes in handy when modeling real-world problems algebraically.