Problem 104 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$5(7-z)+12 z=0 ;-5$$

Step-by-Step Solution

Answer

-5 is not a solution to the equation \(5(7-z)+12 z=0\).

Step 1: Substitute Given Number into the Equation

Substitute \(-5\) for \(z\) in the equation:\[5(7 - (-5)) + 12 (-5) = 0\]

Step 2: Simplify the Equation

This simplifies to \(-60 = 0\]. Both sides of the equation are not equal, hence \(-5\) is not a solution to the equation.

Step 3: Conclude the Solution

-5 does not satisfy the equation \(5(7-z)+12 z=0\), so it is not a solution.

Key Concepts

Solution Verification

Solution verification is crucial in solving algebraic equations as it helps us confirm whether a guessed or given value truly satisfies the equation. In simple terms, verification involves checking if substituting the number back into the equation makes both sides equal.

To verify a solution, take the proposed number and replace the variable in the original equation with this number. Once the substitution is done, calculate both sides of the equation. If both sides are equal, then the number is indeed a solution. If they are not equal, as in this particular problem where \(-60 eq 0\), it means the number is not a solution.

• Verify by substituting the number into the equation.
• Calculate both sides after substitution.
• Check for equality to confirm if the number is a solution.

Substitution Method

Substitution is a fundamental method in algebra used to solve and verify equations. It involves replacing variables with given numbers or expressions to simplify the equation.

In our exercise, substitution was employed when the given number, \(-5\), was substituted into the equation \(5(7-z)+12z=0\). This transforms the equation from one involving a variable into a simple arithmetic operation:\[5(7-(-5))+12(-5) = 0\].

By substituting, we convert the equation into a form that can be easily solved or verified to check if both sides match. The substitution method often marks the first step in assessing whether a given number is a potential solution.

Equation Simplification

Simplifying an equation is all about making it easier to handle by reducing it to the simplest form. This often involves combining like terms, removing brackets, and resolving operations.

During simplification, as seen in our problem, the expression \(5(7-(-5))+12(-5)\) becomes more straightforward:

• First, interpret the sign within the brackets (7 - (-5)) to (7 + 5) = 12.
• Then multiply through: 5(12) = 60.
• Add the second part: 12(-5) = -60.

At the end, the simplified equation becomes \(-60 = 0\), revealing that the left side doesn’t equal the right side, thus verifying that the number -5 is not a solution. Simplification is not just key in making calculations more straightforward, but also in understanding the relationship between different components in an equation.