Problem 20 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(5 x-4=21\)

Step-by-Step Solution

Answer

The solution is \(x = 5\). The equation is not an identity or a contradiction.

Step 1 - Isolate the variable term

To solve the equation, you need to isolate the variable term on one side of the equation. Start by adding 4 to both sides of the equation:\[5x - 4 + 4 = 21 + 4\]This simplifies to:\[5x = 25\]

Step 2 - Solve for the variable

Next, solve for the variable by dividing both sides of the equation by 5:\[\frac{5x}{5} = \frac{25}{5}\]This simplifies to:\[x = 5\]

Step 3 - Check the solution

To verify the solution, substitute the value of the variable back into the original equation. Substitute \(x = 5\) into \(5x - 4 = 21\):\[5(5) - 4 = 21\]\[25 - 4 = 21\]\[21 = 21\]Since both sides of the equation are equal, the solution is correct.

Step 4 - Determine if it's an identity or a contradiction

Since the equation is satisfied with \(x = 5\) and no contradictions are encountered, this equation is neither an identity nor a contradiction. It has a specific solution.

Key Concepts

Variable Isolation

Variable isolation is a key step in solving linear equations. The goal of this step is to get the variable by itself on one side of the equation. This allows you to find its value. To isolate the variable, you use basic algebraic operations such as addition, subtraction, multiplication, and division. For example, in the equation \(5x - 4 = 21\), we start by getting rid of the constant term, \(-4\). We do this by adding 4 to both sides:

• \(5x - 4 + 4 = 21 + 4\)
• This simplifies to \(5x = 25\)

By doing this, we have successfully isolated the term involving the variable, simplifying our equation.

Checking Solutions

After finding a value for the variable, it is crucial to check if this value truly solves the original equation. This step verifies that no mistakes were made. To check the solution of \(x = 5\) for the equation \(5x - 4 = 21\), we substitute \(x = 5\) back into the equation:

• \(5(5) - 4 = 21\)
• This simplifies to \(25 - 4 = 21\)
• Finally, \(21 = 21\)

The left and right sides of the equation are equal, confirming that \(x = 5\) is a correct solution.

Linear Equation Solving Steps

Solving linear equations involves a systematic process of steps designed to find the value of the variable. These steps ensure accuracy and logical progression. Here is the breakdown for the equation \(5x - 4 = 21\):

• Step 1: Isolate the variable term by eliminating constants on the same side as the variable. Add 4 to both sides:
  • \(5x - 4 + 4 = 21 + 4\)
  • Simplify to get \(5x = 25\)
• Step 2: Solve for the variable by performing necessary operations. Here, divide both sides by 5:
  • \(\frac{5x}{5} = \frac{25}{5}\)
  • This simplifies to \(x = 5\)
• Step 3: Check the solution to ensure it satisfies the original equation. Substitute \(x = 5\):
  • \(5(5) - 4 = 21\)
  • Simplify to verify \(21 = 21\)
• Step 4: Determine if the equation is an identity or a contradiction. In this case, the equation is solved by \(x = 5\), which is neither an identity nor a contradiction.

Following these steps helps solve any linear equation reliably.