Problem 31 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 2(x-4)=x+3 $$

Step-by-Step Solution

Answer

The solution is \( x = 11 \).

Step 1: Distribute on the Left Side

First, distribute the 2 on the left side of the equation: \[ 2(x - 4) = 2x - 8 \] So, the equation becomes:\[ 2x - 8 = x + 3 \]

Step 2: Move Variables to One Side

Subtract \( x \) from both sides of the equation to move all the variable terms to one side:\[ 2x - x - 8 = x - x + 3 \]This simplifies to:\[ x - 8 = 3 \]

Step 3: Solve for x

Add 8 to both sides of the equation to solve for \( x \):\[ x - 8 + 8 = 3 + 8 \]Simplifying, we get:\[ x = 11 \]

Step 4: Check the Solution

Substitute \( x = 11 \) back into the original equation to check the solution:Original equation:\[ 2(x - 4) = x + 3 \]Substitute:\[ 2(11 - 4) = 11 + 3 \]Simplify both sides:\[ 2(7) = 14 \] and \[ 14 = 14 \]Both sides of the equation are equal, confirming that \( x = 11 \) is correct.

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that simplifies expressions involving parentheses. It allows us to eliminate parentheses by distributing a factor over the terms inside the parentheses. For example, the expression \( 2(x - 4) \) can be simplified using the distributive property by multiplying 2 with each term inside the parentheses.Here's how it works:

• Multiply the factor outside the parentheses (in this case, 2) by the first term inside the parentheses \((x)\).
• Then, multiply the factor by the second term \((-4)\).

So, \( 2(x - 4) \) becomes \( 2 \times x + 2 \times (-4) \), which simplifies to \( 2x - 8 \). The distributive property is powerful as it transforms expressions into simpler forms, making it easier to solve equations.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra, where the goal is to isolate the variable on one side of the equation. After using the distributive property, we often need to combine like terms and get the variable term on one side of the equation.Following this method:

• First, simplify each side of the equation, if needed. For example, in the equation \( 2x - 8 = x + 3 \), there are no further like terms to combine.
• Next, shift all variable terms to one side of the equation by performing the same operation on both sides. Here, subtract \( x \) from both sides to yield \( x - 8 = 3 \).
• Finally, solve for the variable by isolating it. In this case, add 8 to both sides which results in \( x = 11 \).

This method streamlines the problem-solving process by breaking down the task into manageable steps, making it easier to tackle any linear equation.

Checking Solutions

After solving an equation, it's essential to verify that the solution is correct. Checking solutions helps confirm that no errors were made during calculations and ensures that the solution satisfies the original problem.To check your solution:

• Take the solution you found (here, \( x = 11 \)) and substitute it back into the original equation \( 2(x - 4) = x + 3 \).
• Perform the calculations for both sides. For this equation:
  • Substituting gives \( 2(11 - 4) \) on one side and \( 11 + 3 \) on the other.
• Simplify both: \( 2 \times 7 = 14 \) and \( 14 = 14 \).

Since both sides are equal, our solution \( x = 11 \) is correct. Verification is a crucial part of solving equations as it ensures accuracy and builds confidence in your mathematical abilities.