Problem 65 - Algebra 2

Question

Solve each equation. \(2(a-1)=8 a-6\)

Step-by-Step Solution

Answer

The solution is \(a = \frac{2}{3}\).

Step 1: Distribute the Left Side

Start by distributing the 2 on the left side of the equation to both terms inside the parentheses: \[ 2(a-1) = 2a - 2 \] The equation now becomes: \[ 2a - 2 = 8a - 6 \]

Step 2: Move All Terms Involving 'a' to One Side

Subtract \(2a\) from both sides to gather all the terms with \(a\) on one side: \[ 2a - 2 - 2a = 8a - 6 - 2a \] This simplifies to: \[ -2 = 6a - 6 \]

Step 3: Isolate the 'a' Term

Add 6 to both sides to isolate the term with \(a\): \[ -2 + 6 = 6a \] Simplifying gives: \[ 4 = 6a \]

Step 4: Solve for 'a'

Divide both sides by 6 to solve for \(a\): \[ a = \frac{4}{6} \] Simplify the fraction: \[ a = \frac{2}{3} \]

Step 5: Verify the Solution

Substitute \(a = \frac{2}{3}\) back into the original equation to check the solution:Original equation: \[ 2(a-1) = 8a - 6 \] Substitute \(a = \frac{2}{3}\) into both sides:Left: \[ 2\left(\frac{2}{3} - 1\right) = 2\left(\frac{2}{3} - \frac{3}{3}\right) = 2\left(-\frac{1}{3}\right) = -\frac{2}{3} \] Right: \[ 8\left(\frac{2}{3}\right) - 6 = \frac{16}{3} - \frac{18}{3} = -\frac{2}{3} \]Both sides are equal, so the solution is verified.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify equations. It's an operation where you multiply a single term by each of the terms inside a parenthesis. For instance, in the expression \(2(a-1)\), the distributive property lets you distribute the 2 across both \(a\) and \(-1\), resulting in \(2a - 2\). This process simplifies the expression and prepares it for further steps in solving the equation.

To use the distributive property effectively, remember:

• Multiply the outside number by each term inside the parenthesis.
• Preserve the sign of each term when multiplying, particularly when distributing a negative.
• This property is helpful not only in algebraic equations but also in arithmetic operations, making calculations simpler.

Mastering the distributive property is crucial for tackling more complex algebraic problems.

Solving Equations

When you solve equations, the goal is to find the value of the variable that makes the equation true. This requires a series of logical steps to isolate the variable. One approach is shown in our example equation \(2(a-1) = 8a - 6\).

Here's how we go about it:

• First, simplify both sides of the equation if necessary (using the distributive property or combining like terms).
• Get all terms with the variable on one side of the equation. This often involves adding or subtracting terms from both sides so that all "\(a\)" terms are pooled together.
• Move constant terms to the opposite side. This isolates the variable even further.
• Solve the simplified equation by dividing or multiplying to find the variable's value.

Remember, each action you perform to an equation must be mirrored on both sides to maintain balance. This ensures that you're gradually honing in on the correct value for the variable.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible by reducing it to its lowest terms. This is often a final step in algebraic solutions to make answers neater and easier to interpret.

To simplify \(\frac{4}{6}\) from our solution, follow these steps:

• Determine the greatest common divisor (GCD) of the numerator and the denominator. For 4 and 6, the GCD is 2.
• Divide both the numerator and the denominator by this GCD. So, \(\frac{4}{6}\) becomes \(\frac{4 \div 2}{6 \div 2} = \frac{2}{3}\).

This simple act of division helps reduce the fraction, resulting in a simplified expression that is easier to understand and work with. It's important to simplify fractions whenever possible in your final answer.