Problem 51 - Algebra 1: Concepts and Skills

Question

Solve the equation by simplifying first. $$ -3=a+(-4) $$

Step-by-Step Solution

Answer

The value of \(a\) in the equation is \(1\).

Step 1: Simplification

Start by simplifying the equation. The equation is \(-3=a+(-4)\). In order to get rid of \(-4\) on the right side, we add \(4\) to both sides of the equation. This will bring us a new equation, which is \(-3 + 4 = a + (-4) + 4\). After performing the addition, \(-3+4=1\), and \(a + (-4) + 4=a\), the equation simplifies to \(1=a\).

Step 2: Solving for \(a\)

Our simplified equation from step 1 shows that \(a\) is equal to \(1\). Thus, \(a=1\) is the solution.

Key Concepts

Simplification

Simplification is the process of making an expression or equation easier to understand and solve. In the realm of algebra, we focus on reducing the equation to its simplest form. This ensures an uncomplicated path towards finding a solution. Often, simplification is about eliminating unnecessary elements, thereby de-cluttering the expression or equation.

To illustrate, let's examine our example. We start with the equation \(-3 = a + (-4)\). Here, one of the key steps is recognizing that \(-4\) can be isolated. Simplification will help in making the subsequent steps more apparent and less error-prone.

• Identify terms that can be combined or cancelled out.
• Perform basic arithmetic operations wherever possible.
• Look out for and address any like terms.

In our case, we added 4 to both sides, transforming the equation into \(-3 + 4 = a + (-4) + 4\), which simplifies down to \(1 = a\). This process shown simplifies the expression and ultimately allows us to see the solution clearly.

Addition Property of Equality

The Addition Property of Equality is fundamental in solving equations. It allows us to maintain balance in an equation when adding the same quantity to both sides. This property ensures that the two sides of the equation remain equal even as modifications are made.Consider the equation we begin with, \(-3 = a + (-4)\). Applying the Addition Property, we add 4 to both sides. This adjustment does not change the equation's truth but rather transforms it while preserving equality:

• Initial: \(-3 = a + (-4)\)
• Add 4: \(-3 + 4 = a + (-4) + 4\)
• Resulting: \(1 = a\)

It's like balancing scales; adding or subtracting the same weight from both sides will keep the scales balanced. This property is powerful. It gives us the ability to rewrite equations in forms that are easier to manage and solve.

Algebraic Equations

An algebraic equation is a statement of equality containing one or more variables. It showcases a relationship between known and unknown values we have to solve for. In algebra, solving these equations is akin to solving a puzzle where each piece has its place.In our specific equation, \(-3 = a + (-4)\), the variable is \(a\). Solving algebraic equations generally involves several steps:

• First, we simplify the equation by removing or consolidating terms, just like we did through simplification.
• Then we apply properties such as Addition or Subtraction Property of Equality to isolate the variable.
• The goal is to express the variable clearly, like finding \(a = 1\) in our solved equation.

Once the variable is isolated, you verify your solution by plugging it back into the original equation if needed. This step confirms that both sides of the equation remain equal, thereby supporting that your solution is correct.