Problem 17 - Prealgebra

Question

Solve each equation. Check your solution. $$3(a-1)=4(a-1.5)$$

Step-by-Step Solution

Answer

The solution is \(a = 3\). The solution checks out.

Step 1: Distribute Terms

Start by distributing the terms on both sides of the equation. For the left side, distribute 3 into \(a-1\); for the right side, distribute 4 into \(a-1.5\).Left Side: \[3(a-1) = 3a - 3\]Right Side:\[4(a-1.5) = 4a - 6\]

Step 2: Set the Equations Equal

After distribution, set the two expressions equal to each other. \[3a - 3 = 4a - 6\]

Step 3: Isolate the Variable

To isolate the variable, start by getting all terms with 'a' on one side and numerical terms on the other side.Subtract 3a from both sides:\[-3 = a - 6\]

Step 4: Solve for the Variable

Add 6 to both sides to solve for 'a'.\[3 = a\]So, the solution is \(a = 3\).

Step 5: Check the Solution

Substitute \(a = 3\) back into the original equation to verify.Original equation:\[3(a-1) = 4(a-1.5)\]Substituting \(a = 3\):\[3(3-1) = 4(3-1.5)\]Calculate both sides:\[3 \times 2 = 4 \times 1.5\]\[6 = 6\]The solution checks out, so \(a = 3\) is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify and solve equations. It involves distributing a multiplier across a set of terms within parentheses. Think of it like sharing something equally among all those terms. In prealgebra exercises, the distributive property will often look like this:

• Take a number outside the parentheses and multiply it with each term inside the parentheses.
• This ensures that every part of the equation is accounted for.

In our exercise, the distributive process looks like this:

• Left Side: Distribute 3 into \(a-1\), giving you \(3a - 3\).
• Right Side: Distribute 4 into \(a-1.5\), resulting in \(4a - 6\).

Once the terms have been distributed, the equation becomes easier to manage and solve.

Isolate the Variable

To solve equations, particularly in prealgebra, you often need to isolate the variable. In simpler terms, this means getting the variable by itself on one side of the equation. This helps you to identify what value the variable holds. Follow these steps:

• First, arrange the equation so that all terms containing the variable are on one side. Usually, this is achieved through addition or subtraction.
• Next, perform similar operations to the constants to remove them from the side with the variable.

In the provided example, we subtract \(3a\) from both sides to shift all the 'a' terms to one side:

• Equation becomes: \(-3 = a - 6\).\
• Then, by adding 6 to both sides, you further simplify: \(3 = a\).\

Voila! The variable is isolated, and you have found that \(a = 3\).