Problem 24 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(2 x+4-x=4 x-5\)

Step-by-Step Solution

Answer

The solution is \(x = 3\) and the equation is neither an identity nor a contradiction.

Step 1: Combine Like Terms

Combine the like terms on both sides of the equation. On the left side, combine the terms with 'x'.

Step 2: Simplify Each Side

Simplify the left side of the equation. This gives us: \[2x + 4 - x = 4x - 5\] which simplifies to: \[x + 4 = 4x - 5\]

Step 3: Isolate the Variable

Subtract 'x' from both sides to start isolating the variable: \[x + 4 - x = 4x - 5 - x\] which simplifies to: \[4 = 3x - 5\]

Step 4: Solve for x

Next, isolate 'x' by adding 5 to both sides: \[4 + 5 = 3x - 5 + 5\] which simplifies to: \[9 = 3x\] Now divide by 3: \[\frac{9}{3} = \frac{3x}{3}\] which simplifies to: \[x = 3\]

Step 5: Check the Solution

Substitute 'x = 3' back into the original equation to verify: \[2(3) + 4 - 3 = 4(3) - 5\] which simplifies to: \[6 + 4 - 3 = 12 - 5\] This simplifies to: \[7 = 7\] Since both sides are equal, the solution is correct.

Step 6: Determine Type of Equation

Since the solution works and there are no contradictions, this equation is a regular equation, neither an identity nor a contradiction.

Key Concepts

Combining Like Terms

When solving linear equations like the one in our exercise \(2x + 4 - x = 4x - 5\), the first step is to combine like terms. Combining like terms simplifies the equation and makes it easier to solve. Here's what you need to do:

• Identify terms that have the same variable. For example, on the left side of our equation, both \(2x\) and \(-x\) have the variable 'x'.
• Add or subtract these terms to combine them. In our case, \(2x - x\) simplifies to \(x\).

This gives us the new, simpler equation: \(x + 4 = 4x - 5\). By combining like terms, we reduce the complexity and get closer to isolating the variable.

Isolating the Variable

The next key step in solving linear equations is isolating the variable. This means getting the variable, usually 'x', alone on one side of the equation. To achieve this, follow these steps:

• Move variable terms to one side of the equation and constant terms to the other. For the equation \(x + 4 = 4x - 5\), you can subtract 'x' from both sides: \(4 = 3x - 5\).
• Next, eliminate the constant term on the side with the variable by adding or subtracting it from both sides. Adding 5 to both sides of our equation, we get \(4 + 5 = 3x - 5 + 5\), which simplifies to \(9 = 3x\).
• Finally, divide or multiply to get the variable by itself. Dividing both sides by 3 gives us \(x = 3\).

Through this process, we isolate 'x' and find that \(x = 3\). Isolating the variable is crucial for arriving at a simplified and solvable form of the equation.

Checking Solutions

Once we have found a solution, it is important to verify that it is correct by checking the solution. Plug the solution back into the original equation:

• Substitute the found value of 'x' back into the original equation. For our example, we substitute \(x = 3\) back into \(2x + 4 - x = 4x - 5\).
• Perform the calculations to check if both sides of the equation are equal. Substituting 3 for 'x', we get \(2(3) + 4 - 3 = 4(3) - 5\), which simplifies to \(6 + 4 - 3 = 12 - 5\) and further simplifies to \(7 = 7\).
• If both sides are equal, the solution is verified as correct.

In our case, the solution \(x = 3\) makes both sides of the equation equal, confirming it is correct. Checking solutions ensures that the solution found truly satisfies the original equation.