Problem 73 - Algebra 1

Question

Solve the equation. $$ \frac{1}{2} a+8 \frac{1}{2} a=3 $$

Step-by-Step Solution

Answer

The solution for the equation is \(a = 0.67\).

Step 1: Combine like terms

As both terms have 'a', combine the terms on left hand side of the equation: \(\frac{1}{2}a+8 \times \frac{1}{2}a=3\). Doing so, we have \(4.5a = 3\).

Step 2: Solve for 'a'

Dividing both sides of the equation by 4.5 to solve for 'a', gives: \(a = \frac{3}{4.5}\) or \(a = 0.67\)

Key Concepts

Combining Like Terms

Combining like terms is a fundamental concept when solving linear equations.
Like terms are terms in an equation that have the same variable raised to the same power.
They can be combined by adding or subtracting their coefficients. In this context, the objective is to simplify the equation to make it easier to solve.In our original exercise, the equation is \( \frac{1}{2} a + 8 \cdot \frac{1}{2} a = 3 \).
Both terms on the left-hand side contain the variable 'a'.
To combine them, add their coefficients. This means:

• The first term's coefficient is \( \frac{1}{2} \).
• The second term's coefficient becomes \( 8 \times \frac{1}{2} = 4 \).
• Add the coefficients: \( \frac{1}{2} + 4 = 4.5 \).

Thus, combining like terms results in \( 4.5a = 3 \).
Simplifying the equation by combining like terms is an important step before diving into further calculations.

Arithmetic Operations

Arithmetic operations involve addition, subtraction, multiplication, and division.
These operations are used to manipulate numbers and terms in an equation and are crucial in solving linear equations. To solve for an unknown, we often use these operations to isolate the variable on one side of the equation.In the solution process:
The essential arithmetic operation used is division, to isolate the variable.
After combining like terms, we reach \( 4.5a = 3 \).
To solve for 'a', divide both sides by 4.5. This operation gives:

• Divide the right side: \( \frac{3}{4.5} \).
• This fraction simplifies to approximately \( 0.67 \).

Performing arithmetic operations correctly ensures that the solution is accurate.
It is vital to maintain balance in the equation by applying the same operation to both sides.

Fractions in Equations

Handling fractions can sometimes be tricky, but they follow the same basic rules as whole numbers.
In linear equations, fractions might appear as coefficients or constants. They require careful arithmetic handling to simplify the equation and solve for the variable efficiently.In the given exercise, fractions play a role in:

• The coefficient of 'a', \( \frac{1}{2} \).
• The process of dividing the equation by 4.5 to solve for 'a'.

When dealing with fractions in equations:

• Consider converting fractions to decimals for easier computation, as done when \( \frac{1}{2} \) was multiplied by 8.
• Ensure that when dividing, the fraction is correctly simplified: \( \frac{3}{4.5} \approx 0.67 \).
• Always check if the fraction can be reduced or interpreted in a simpler form for clarity.

Fractions might seem overwhelming at first, but with practice, they become as manageable as any other number type.
Remembering these tips will help you efficiently solve equations involving fractions.