Problem 13 - Algebra 1: Concepts and Skills

Question

Solve the equation if possible. Determine whether the equation has one solution, no solution, or is an identity. $$ x-2 x+3=3-x $$

Step-by-Step Solution

Answer

The given equation is an identity as upon simplification, both sides of the equation are identical. Hence, it has infinitely many solutions.

Step 1: Simplification

First, simplify the expression on the left side of the equation: \(-2x + 3\). This simplifies to \(3 - x\). Now the equation can be rewritten as \(3 - x = 3 - x\).

Step 2: Comparison

As observed, both sides of the equation are identical, indicating an identity.

Key Concepts

Equation Simplification

Equation simplification is a fundamental process in algebra which involves altering an equation to a simpler or more efficient form without changing its solution. This generally involves combining like terms, reducing fractions, and factoring, among other methods. Simplification can make equations more manageable and can sometimes help reveal solutions or special properties of the equations, such as identities or inconsistencies.

For example, when faced with an equation like \( x - 2x + 3 \), simplifying it involves combining like terms—terms that have the same variable raised to the same power. Here, \(-2x\) and \(+x\) are like terms, and when combined, they give us \(-x\). So the equation simplifies to \(-x + 3\), which can further be written as \(3 - x\) for clarity. Simplifying equations is an essential skill as it paves the way for more complex operations such as solving the equation.

Identities in Algebra

Identities in algebra are equations that are true for every value of the variable involved. They depict an intrinsic property of numbers and can often be recognized by simplifying an equation to discover the same expression on both sides. This indicates that no matter what value you substitute for the variable, the equation will always hold true.

An example might look like \(a + b = b + a\), which demonstrates the commutative property of addition—an identity because it is always valid regardless of the values of \a\ and \b\. In the context of our exercise, once the equation \( x - 2x + 3 = 3 - x \) is simplified on both sides, we are left with \( 3 - x \) on both sides of the equation. This means you're looking at an identity because this equation is true for every value of \x\ you could choose.

No Solution Equations

Equations that have no solution are those in which no value of the variable will satisfy the equation; they typically result from attempting to equate two expressions that cannot be made identical. These types of equations are sometimes referred to as contradictions, as they propose a condition that is inherently unsatisfiable.

For instance, an equation like \(5x + 2 = 5x + 3\) cannot be true for any real value of \x\, because no matter what \x\ you substitute, the two sides will never be equal due to the differing constants (2 and 3). This would be categorized as a 'no solution' scenario. However, in the original exercise \(x - 2x + 3 = 3 - x\), although it may appear at first glance that it might not have a solution, once simplified, reveals itself to be an identity and therefore has an infinite number of solutions. Recognizing no solution equations is crucial in understanding the range of possibilities that equations can present and when to stop pursuing a solution that doesn't exist.