Problem 17 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 7 and 8 . \(2 x-4=16\)

Step-by-Step Solution

Answer

The solution is \(x = 10\).

Step 1: Isolate the Variable

Begin by adding 4 to both sides of the equation to get the variable term by itself:\[2x - 4 + 4 = 16 + 4\]This simplifies to:\[2x = 20\]

Step 2: Solve for x

Now divide both sides of the equation by 2 to solve for \(x\):\[\frac{2x}{2} = \frac{20}{2}\]This simplifies to:\[x = 10\]

Step 3: Check the Solution

Substitute \(x = 10\) back into the original equation to verify:\[2(10) - 4 = 16\]Calculate:\[20 - 4 = 16\]Since the left side of the equation equals the right side, the solution \(x = 10\) is correct.

Key Concepts

Solving Equations

When faced with a linear equation like \(2x - 4 = 16\), the primary goal is to find the value of \(x\) that makes the equation true. In mathematics, this process is known as solving an equation. To solve an equation means to find all values of the variable that satisfy the equation. This often involves a series of steps to manipulate the equation and isolate the variable. By understanding the structure of linear equations, you can apply strategies like combining like terms, using inverse operations, and simplifying expressions to efficiently solve the equation. Here are some general steps for solving linear equations:

• Identify the equation to be solved.
• Use inverse operations to simplify the equation.
• Rearrange the equation to isolate the variable.
• Perform any necessary arithmetic to find the variable's value.
Each step fits into a larger methodical process, helping to systematically solve and verify solutions.

Isolation of Variable

Isolation of the variable is crucial in solving equations, as it helps simplify the process by focusing only on the variable in question. In our example, the equation \(2x - 4 = 16\) requires isolating the variable \(x\). This involves rearranging the equation so that \(x\) stands alone on one side of the equation.The process can be broken down into smaller steps:

• Identify and eliminate any constants or coefficients that are attached to the variable.
• Use inverse operations to move these numbers to the other side of the equation.
• Simplify the equation step-by-step until the variable is isolated.

For instance, adding \(4\) to both sides eliminates the \(-4\) and using division by \(2\) takes care of the coefficient of \(x\). This leaves \(x\) by itself, allowing us to plainly see its value.

Checking Solutions

Once a potential solution is obtained, it's important to verify it. This ensures that your solution is not only reasonable but correct. In mathematics, checking solutions involves substituting the found value back into the original equation to ensure both sides remain equal.Consider our earlier example with the solution \(x = 10\):

• Substitute \(x = 10\) back into the original equation \(2x - 4 = 16\).
• Calculate the left-hand side: \(2(10) - 4\) which simplifies to \(20 - 4\).
• Verify equality by ensuring the result \(16\) matches the right-hand side of the equation.

The equality of both sides confirms the solution is accurate. Checking solutions is a crucial final step in solving equations, reinforcing the correctness and reliability of the answer.