Problem 74 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

No, 1 is not a solution of the equation.

Step 1: Substitute the given number into the equation

We need to check if the given number, 1, is a solution to the equation. To do this, substitute 1 for the variable \(x\) in the equation \(4 = 1 - x\). This gives us the equation \(4 = 1 - 1\).

Step 2: Simplify the equation

Now simplify the right-hand side of the equation. We have \(1 - 1\) which equals 0. So, the equation becomes \(4 = 0\).

Step 3: Verify if the equation holds true

Compare both sides of the equation after simplification. We have \(4 = 0\), which is not true.

Step 4: Conclude whether the number is a solution

Since substituting 1 for \(x\) does not satisfy the equation \(4 = 1 - x\), 1 is not a solution of the equation.

Key Concepts

Equation Substitution

Equation substitution is an essential step when determining if a given number is a solution to an equation. It involves replacing the variable in the equation with the specified number that we want to test. In our example, the equation given is \(4 = 1 - x\), and we are asked to check if the number 1 is a solution. This means we substitute 1 in place of \(x\).

After the substitution, the equation becomes \(4 = 1 - 1\). This step does not yet tell us if the number is a solution; it's simply transforming the original algebraic expression into a more numerical one. It's crucial because it sets the stage for simplifying the equation to see if both sides can be made equal with that substitution.

Simplifying Equations

Simplifying equations is usually the next step after substitution. It involves performing basic arithmetic to simplify the equation. From our example equation, after substituting 1 for \(x\), the equation becomes \(4 = 1 - 1\).

Now simplify the right side of the equation. Calculate \(1 - 1\), which equals 0. So, the simplified equation is \(4 = 0\).

This step is vital because it allows us to see whether the equation holds true after substituting the number. If both sides of the simplified equation are equal, the number is a solution. In this case, however, they are not equal (\(4\) is not the same as \(0\)).

Verification of Solutions

Once equation substitution and simplification are done, the last step is verifying if the original equation holds true with the substituted number. This involves comparing both sides of the equation after simplification.

In our example, after substituting and simplifying, we got \(4 = 0\). The verification step is asking the question: "Is this true?" Since 4 is not equal to 0, the conclusion is that 1 is not a solution to the equation \(4 = 1 - x\).

Verification of solutions is critical as it confirms or refutes the potential solution. If the two sides don't match, like in this case, the number isn't a solution. This step helps in ensuring that the calculations and logic used in previous steps were correctly applied.