Problem 43 - 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. $$ 8+6 a=6 a-1 $$

Step-by-Step Solution

Answer

The given equation has no solutions.

Step 1: Simplify the Equation

The first thing to do is to simplify the given equation by combining like terms. This means it's necessary to bring all variables on one side and constants on the others. In this case it means subtracting \(6a\) from both sides of the equation \(8+6 a-6 a=6 a-6 a-1\), which simplifies to \(8 = -1\)

Step 2: Analyze the Result

Upon simplifying the equation, it results in \(8=-1\), which is not a true statement. Therefore, this indicates that the equation has no real solutions.

Key Concepts

One Solution

When we say a linear equation has one solution, we're indicating that there's a single unique value for the variable that makes the equation true. This situation occurs when the variable, let's call it 'x', appears on one or both sides of the equation, and all of our algebraic manipulations lead to a valid statement like \( x = 5 \). This means if we substitute 5 for 'x' into the original equation, both sides would hold equal values, verifying the solution as correct.

No Solution

In contrast, when an equation has no solution, this means there's no possible value for the variable that makes the equation true. During simplification, these equations typically result in a statement that's always false, like \( 8 = -1 \). No matter what number we substitute for the variable, the equation will never hold true. This is also referred to as an inconsistent equation since it doesn't consistent with any real number.

Identities in Algebra

Moving over to identities in algebra, these are equations that are true for all values of the variables involved. A simplified algebraic identity looks something like \( 3x + 2 = 3x + 2 \). No matter what value 'x' takes, both sides of the equation are inherently equal, hence true for every possible 'x'. An identity suggests an infinite number of solutions since the equation does not change the relationship between the variables.

Combining Like Terms

To solve equations efficiently, combining like terms is a critical step. This means adding or subtracting terms with the same variable and exponent. Consider an equation like \( 2x + 3 + 4x - 2 \); by combining like terms (here, terms involving 'x'), we can simplify this to \( 6x + 1 \). This step reduces complexity and sets us up for clearer, more straightforward solutions.

Simplified Equation

A simplified equation is the result after using various algebraic operations to reduce an equation to its most basic form without changing the original meaning. Simplifying often involves combining like terms, applying the distributive property, and removing any redundant parts of the equation. A simplified equation presents a cleared picture, whether it’s finding one solution, no solution, or identifying it as an identity.