Problem 88 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

5 is not a solution because it leads to a false statement (4 ≠ -4).

Step 1: Identify the Equation

The equation provided is \( 4 = 1 - x \). We need to determine if \( x = 5 \) makes the equation true.

Step 2: Substitute the Value into the Equation

Replace \( x \) with 5 in the equation. The equation becomes \( 4 = 1 - 5 \).

Step 3: Simplify the Right Side of the Equation

Calculate the right-hand side where \( 1 - 5 = -4 \).

Step 4: Compare Both Sides of the Equation

The simplified equation is \( 4 = -4 \). Since 4 does not equal -4, the equation is false.

Step 5: Conclusion

Since substituting \( x = 5 \) into the equation gives \( 4 = -4 \) which is false, 5 is not a solution of the equation \( 4 = 1 - x \).

Key Concepts

Solution Verification

Verifying a solution in algebra means checking if a particular value satisfies an equation. To verify if a number is a solution, we substitute the number into the equation in place of the variable, then check if both sides of the equation equal.
In our exercise, we replaced the variable \( x \) with the number 5 into the equation \( 4 = 1 - x \). After substitution, the equation transformed to \( 4 = 1 - 5 \). When simplified, \( 1 - 5 \) results in \(-4\). Clearly, 4 does not equal -4, indicating that 5 is not a valid solution.
Remember, for a number to be a solution, it must make the equation true by equating both sides. If the substitution leads to a false statement (like 4 = -4), the number is not a solution. This step-by-step approach helps ensure accuracy and clarity.

Equation Solving

Solving an algebraic equation means finding the value of the unknown variable that makes the equation true.
Generally, solving an equation involves isolating the variable on one side of the equation. This often requires performing inverse operations like addition, subtraction, multiplication, or division.
In our example, the equation was \(4 = 1 - x\). The goal was to determine if substituting \(x = 5\) results in a true statement. Upon substitution, the equation becomes \(4 = 1 - 5\), which simplifies to \(4 = -4\). This is false, showing that 5 does not solve the equation. Understanding the process of solving equations also involves checking your work by plugging the solution back into the original equation. This ensures the calculated solution actually satisfies the equation.

Substitution Method

The substitution method is a straightforward algebraic technique used to verify or find the solution of an equation. It involves replacing a variable with a specific numerical value. This method is particularly useful when you are given a potential solution and need to determine its validity.
In our case, the substitution was done by replacing \( x \) with 5 in the equation \(4 = 1 - x\). This resulted in the equation \(4 = 1 - 5\). The next step is to simplify the side where substitution was made, producing the expression \(4 = -4\).
If after substitution the equation holds true, the number is a solution. If not, as was the case here, then the number is not a solution. This method provides a systematic way to test potential solutions and reinforces understanding of equation concepts by immediately revealing discrepancies.