Problem 23 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-4 a+3=-9$$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 3\).

Step 1: Identify the equation

The given equation is:\[-4a + 3 = -9\]

Step 2: Apply the Addition Property of Equality

To isolate the term with the variable \(a\), we can start by getting rid of the constant on the left side of the equation. Subtract 3 from both sides:\[-4a + 3 - 3 = -9 - 3\]Simplifying both sides, we get:\[-4a = -12\]

Step 3: Solve for the variable

Now that we have the equation \(-4a = -12\), we can solve for \(a\) by dividing both sides by \(-4\):\[-4a / -4 = -12 / -4\]This simplifies to:\[a = 3\]

Key Concepts

Addition Property of Equality

The **Addition Property of Equality** is a fundamental concept in algebra. It's a simple yet powerful tool used in solving equations. The idea is straightforward: you can add the same value to both sides of an equation without changing the equality. This helps in isolating terms and simplifying the equations in order to find the solution.

• **How it works**: If you have an equation like \(x + 3 = 7\), you add or subtract a number to both sides to maintain balance. For instance, subtracting 3 from both sides results in \(x = 4\).
• **Why it's used**: It helps to move constants or terms from one side to the other. It's often the first step in isolating a variable.This concept helps to make equations easier to manage and solve.

By keeping the balance, the Addition Property of Equality ensures the equation remains valid, allowing you to manipulate and solve for the unknown variable.

Solving Equations

When solving equations, the goal is to find the value of the unknown variable that makes the equality true. In other words, we are looking for a number that can replace the variable to make both sides of the equation equal. Solving equations requires a clear understanding and deliberate action using algebraic properties.

• **Steps to Solve**: Identify the equation and apply algebraic manipulations using properties such as addition, subtraction, multiplication, and division to unravel the equation.
• **Example**: Let's take the equation \(-4a + 3 = -9\). The initial step is using the addition property to remove constants from one side (here, subtracting 3 from both sides gives \(-4a = -12\)).
• **Checking Solutions**: Once you find a solution (e.g., \(a = 3\)), substitute it back into the original equation to verify it satisfies the equation.

Algebraic Manipulation

Algebraic Manipulation is the use of mathematical operations and properties to artfully transform and solve equations. The goal here is to rearrange the equation to isolate the variable of interest.

• **Simplification**: Reduce equations to simpler forms using addition, subtraction, multiplication, and division. This is often accompanied by combining like terms.For instance, simplifying \(-4a + 3 - 3 = -9 -3\) leads us to \(-4a = -12\).
• **Isolating the Variable**: This involves getting the variable on one side of the equation, often leading to a direct solution. In our example, dividing both sides of \(-4a = -12\) by \(-4\) isolates \(a\).
• **Order of Operations**: Follow the order of operations (PEMDAS/BODMAS) while manipulating equations to avoid mistakes.

The mastery of algebraic manipulation allows one to solve increasingly complex equations by breaking them down into manageable steps.