Problem 10 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ a-0=5 $$

Step-by-Step Solution

Answer

Yes, \(a = 5\) is a solution of the equation \(a - 0 = 5\).

Step 1: Substitute the given number into the equation

The first step involves substituting \(a = 5\) into the equation \(a - 0 = 5\). In doing so, we obtain \(5 - 0 = 5\).

Step 2: Check for equality

Next, we must check if both sides of the equation are equal. In our case, we're checking if \(5 - 0 = 5\). Once we perform the subtraction on the left side, we obtain \(5 = 5\).

Step 3: Determine if the number is a solution

Finally, since both sides of the equation are equal, we can conclude that the given number, \(a = 5\), is indeed a solution of the equation \(a - 0 = 5\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in algebra for solving equations, especially systems of linear equations. It provides a straightforward approach to finding an unknown in terms of other known values. In the context of our exercise, substitution involves replacing the variable, in this case, 'a', with its given value of 5. Here's how it looks in action:

Original Equation: \( a - 0 = 5 \)
Substitute 'a' with 5: \( 5 - 0 = 5 \)
The aim here is to simplify the equation using this value, ultimately making it possible to validate whether the given value satisfies the equation. This process can be particularly helpful in more complex scenarios where multiple equations and variables are involved, allowing for the progressive simplification of a problem. When teaching the substitution method, encourage students to practice with various types of equations, reinforcing the concept through repetition.

Equality in Algebra

At the heart of algebra lies the concept of equality, symbolized by the equal sign '='. It signifies that the values on either side of the equation balance each other out. Understanding this principle is crucial for mastering algebraic solutions. The equation from our exercise, \( a - 0 = 5 \) at first glance appears simple, but it's a perfect illustration of the equality concept.

When we substitute 'a' with 5, we get \( 5 - 0 = 5 \), which confirms that both sides are indeed equal, satisfying the equation. It's essential to emphasize to students that equations can be like scales; just as weights added or removed from one side of a scale must be countered by the same on the other side to keep it balanced, mathematical operations done to one side of an equation must be mirrored on the other.

Algebraic Solutions

Algebraic solutions involve finding values for variables that make a given equation true. In our example, the algebraic solution to the equation \( a - 0 = 5 \) is the value of 'a' that makes the equation hold—namely, \( a = 5 \). After substituting and establishing equality, one might deem this as a valid solution, completing the process.

However, beyond simple equations like this, the process of finding algebraic solutions can get complex, involving operations such as factoring, rewriting equations, and more. It's critical to practice various techniques and types of equations, as this paves the way for solving real-world problems where variables and equations represent physical quantities. Always verify your solutions by substituting them back into the original equations to ensure accuracy.