Problem 44 - Algebra 1: Concepts and Skills

Question

Without writing the steps of a solution, determine whether the equation has one solution, no solution, or is an identity. $$ 6 a+8=2 a $$

Step-by-Step Solution

Answer

The given equation has one solution, a = -2.

Step 1: Simplify the Equation

We start by subtracting \(2 a\) from each side of the equation, which will give a new equation which is \(6 a - 2 a + 8 = 0\). Further simplifying, we obtain \(4 a + 8 = 0\).

Step 2: Isolate for variable a

Next, let's isolate the variable a, by subtracting 8 from each side of the equation. This gives us \(4 a = -8\). Then dividing each side of the equation by 4 will give us the solution for variable a. Thus, \( a = -2\)

Step 3: Determine the nature of the equation

We have simplified the equation and found the value of the variable a. Thus, this equation has one solution. It is not an equation with no solution or an identity.

Key Concepts

Algebraic Identities

Algebraic identities are mathematical expressions that are true for all values of the variables involved. They are used frequently to simplify algebraic expressions and solve equations. For example, the basic binomial identity \( (x+y)^2 = x^2 + 2xy + y^2 \) is true for any values of x and y.

Understanding algebraic identities can provide powerful tools for analyzing and solving equations. In the context of the given exercise, recognizing that no algebraic identity applies simplifies the process of determining the nature of the equation, as you're not misled into thinking that the equation is true for all values of the variable 'a'.

No Solution Equations

Equations with no solution are those for which no value of the variable will satisfy the equation. They usually occur when a simplification leads to a contradiction, such as \( 0 = 3 \). In such cases, the equation is found to hold no truth for any value, and is deemed unsolvable or inconsistent.

In the exercise, one might initially consider if the given equation could lead to a contradiction, indicating no solution. Through simplification, we discover that is not the case for this particular example, as we are able to isolate the variable and find a specific solution, \( a = -2 \).

Isolating Variables

Isolating the variable is a standard method in solving linear equations and a critical skill in algebra. The goal is to get the variable on one side of the equation with a coefficient of 1. This process often involves operations such as addition, subtraction, multiplication, and division.

For the given problem, isolating the variable is done by eliminating like terms and constants from one side of the equation to simplify it to the form \( a = \text{some number} \). Steps taken include subtracting \( 2a \) and \( 8 \) from both sides, and finally dividing by \( 4 \) to solve for 'a'. The result \( a = -2 \) means we have isolated 'a' and found the unique solution to the equation.