Problem 45 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-[2 x-(5 x+2)]=2+(2 x+7)\)

Step-by-Step Solution

Answer

\(x = 7\)

Step 1: Simplify both sides

Left side: \(-[2x-(5x+2)] = -[2x-5x-2] = -[-3x-2] = 3x+2\)
Right side: \(2+(2x+7) = 2x+9\)

Step 2: Solve the equation

\(3x+2 = 2x+9\)
\(3x-2x = 9-2\)
\(x = 7\)

Step 3: Check

Left: \(-[2(7)-(5(7)+2)] = -[14-37] = -[-23] = 23\)
Right: \(2+(2(7)+7) = 2+21 = 23\) \(\checkmark\)
The solution is \(x = 7\).

Key Concepts

Distributing Negative Sign

When solving linear equations, it's crucial to distribute any negative signs across the terms inside the parentheses correctly. In our exercise, the equation is \(-[2x-(5x+2)]=2+(2x+7)\). We need to distribute the negative sign to both terms inside the brackets. Here's how:

• First, look at what's inside the brackets: \(2x - (5x + 2)\).
• Then, distribute the negative sign to \(5x\) and \(2\): \(-2x + 5x + 2\).
• Now, simplify it to: \(3x + 2\).

Distributing the negative sign is like breaking it down into elementary steps, making the rest of the equation-solving process smoother. This is an essential step to ensure accuracy.

Isolating Variable

Isolating the variable is the next important step in solving linear equations. Once we have simplified our equation to \(3x + 2 = 2x + 9\), follow these steps:

• Subtract \(2x\) from both sides to get all the \(x\)-terms on one side: \(3x + 2 - 2x = 2x + 9 - 2x\).
• This will simplify to: \(x + 2 = 9\).
• Next, remove the constant from the left side by subtracting \(2\) from both sides: \(x + 2 - 2 = 9 - 2\).
• The result is \(x = 7\).

Isolating the variable involves rearranging the equation to solve for one instance of the variable. Every step ensures we correctly handle each term, leading us to the solution.

Checking Solutions

Once we have solved for the variable, we need to verify our solution by substituting it back into the original equation. This ensures our solution is correct.

• Start with the original equation: \(-[2x-(5x+2)]=2+(2x+7)\).
• Substitute \(x = 7\) into the equation: \(-[2(7)-(5(7)+2)]=2+(2(7)+7)\).
• Simplify both sides: \(-[14- (35+2)]=2+(14+7) \).
• Continue simplifying: \(-[14-37] = 2+21\), which simplifies to: \(-(-23) = 23\).
• Finally, this simplifies to: \(23 = 23\), proving our solution is correct.

Checking solutions helps confirm that every step taken was accurate, solidifying the correctness of our solution.