Problem 24 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-5 a+10=50$$

Step-by-Step Solution

Answer

The solution is \(a = -8\).

Step 1: Identify the Equation

The given equation is \(-5a + 10 = 50\). This equation needs to be solved for \(a\).

Step 2: Apply the Addition Property of Equality

Subtract 10 from both sides of the equation to eliminate the constant term on the left side. This gives us:\(-5a + 10 - 10 = 50 - 10\).

Step 3: Simplify the Equation

The equation simplifies to:\(-5a = 40\).

Step 4: Use the Division Property of Equality

Divide each side by \(-5\) to solve for \(a\): \(a = \frac{40}{-5}\).

Step 5: Simplify the Solution

Perform the division to find the value of \(a\):\(a = -8\).

Key Concepts

Addition Property of Equality

In algebra, the addition property of equality is a fundamental concept. It states that adding or subtracting the same number from both sides of an equation keeps the equality valid.
For instance, when we have the equation \(-5a + 10 = 50\), our goal is to isolate the variable, which in this case is \(a\).
To do that, we need to get rid of the constant \(10\) on the left side of the equation. By using the addition property of equality, we subtract \(10\) from both sides.

• The equation becomes \(-5a + 10 - 10 = 50 - 10\).
• This simplifies the equation to \(-5a = 40\).

This method helps us keep the equation balanced while moving terms around to solve for the variable.

Division Property of Equality

After simplifying the equation using the addition property of equality, a variable is often multiplied by a number. The division property of equality helps us solve the equation by dividing each side by that number. This technique ensures the equation remains true.
In our example, after simplifying, we have the equation \(-5a = 40\).
To solve for \(a\), we need to get rid of the negative coefficient \(-5\). Here's how it's done:

• Divide each side of the equation by \(-5\): \(\frac{-5a}{-5} = \frac{40}{-5}\).
• This gives us the simplified form: \(a = -8\).

The division property of equality allows us to isolate the variable while maintaining the equation's balance.

Step-by-Step Algebra Solutions

Step-by-step algebra solutions provide a clear and systematic way to solve equations. This method is particularly helpful for understanding and mastering algebra concepts.
Let's break down the problem-solving process using our equation \(-5a + 10 = 50\):

• Identify the equation: Recognize it and understand what you need to solve for, which in this case is \(a\).
• Apply the addition property: Subtract \(10\) from both sides to simplify the equation (\(-5a = 40\)).
• Simplify using division: Divide each side by \(-5\) to solve for \(a\), resulting in \(a = -8\).

By following a structured approach, complex equations become more manageable. This technique fosters a deeper understanding and builds confidence in solving algebraic equations.