Problem 74 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. \(4=1-x ; 5\)

Step-by-Step Solution

Answer

5 is not a solution because it doesn't satisfy the equation.

Step 1: Substitute the Value

To determine if 5 is a solution to the equation \(4 = 1 - x\), substitute 5 for \(x\) in the equation. The equation becomes: \(4 = 1 - 5\).

Step 2: Simplify the Equation

Simplify the right-hand side of the equation \(1 - 5\). Performing the subtraction gives you \(-4\), turning the equation into \(4 = -4\).

Step 3: Compare Both Sides

Compare the results on both sides of the equation. You have \(4\) on the left and \(-4\) on the right. Since \(4 eq -4\), the number 5 is not a solution.

Key Concepts

Substitution Method

When we solve equations, especially when checking if a number is a solution, the substitution method is our go-to technique. It's straightforward and involves a simple replacement process. Think of it like checking to see if a puzzle piece fits into a given puzzle space.

Here's what you do:

• Take the number you suspect might be a solution and substitute it for the variable in the equation. Here, our variable is \(x\).
• If our equation is \(4 = 1 - x\) and we want to check if \(5\) is a solution, we replace \(x\) with \(5\).

After substituting, the equation becomes \(4 = 1 - 5\). This step lets us "test" if our guess (in this case, \(5\)) works in the equation.

It's like testing out whether your key fits and unlocks the door!

Simplifying Equations

Simplifying equations is like tidying up a messy room; we want everything to be in its simplest form. After using the substitution method, we often need to simplify to make sense of the equation.

In our equation \(4 = 1 - 5\):

• Look at the right side, which is \(1 - 5\).
• Do the math; subtracting gives us \(-4\).

Now, the equation looks like this: \(4 = -4\).

By simplifying, we can easily compare the numbers on each side. This comparison helps us see clearly whether both sides are equal, confirming if our substituted number is a solution or not.

Solution Verification

After substituting and simplifying, the final step is solution verification.

This is where we put on our detective hat and check if both sides of the equation match. For our simplified equation \(4 = -4\):

• Evaluate both sides. The left side is \(4\), and the simplified right side is \(-4\).
• Since \(4 eq -4\), we can conclude that they are not equal.

If both sides were equal, our number would be a solution. But here, they're not. Thus, \(5\) is not a solution for the equation \(4 = 1 - x\).

Verification is crucial because it confirms our work is accurate and the solution is indeed correct or incorrect. Think of it like double-checking your answers on a test to ensure they're right.