Problem 9 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ -12 x+23=-14 ; \quad x=116 $$

Step-by-Step Solution

Answer

No, \(x = 116\) is not a solution to the equation.

Step 1: Substitute x into the Equation

We need to check if the given value \(x = 116\) satisfies the equation \(-12x + 23 = -14\). Start by substituting \(116\) for \(x\) in the equation:\[-12(116) + 23\].

Step 2: Perform the Multiplication

Calculate the value of \(-12 \times 116\). This is done by multiplying \(-12\) by \(116\):\[ -12 \times 116 = -1392 \].

Step 3: Add the Result to 23

Now, add \(-1392\) to \(23\) to see if the equation is satisfied:\[ -1392 + 23 = -1369 \].

Step 4: Compare the Result to -14

The equation is satisfied if \(-1369\) equals \(-14\). Clearly:\[ -1369 eq -14 \].

Step 5: Conclusion

Since \(-1369\) does not equal \(-14\), the value \(x = 116\) is not a solution to the equation \(-12x + 23 = -14\).

Key Concepts

Substitution Method

The substitution method is a crucial concept in solving linear equations. It involves replacing a variable with a specific value to verify if the equation holds true. When we have an equation like \(-12x + 23 = -14\), we can use this method to check if a given value, say \(x = 116\), is a solution.

• First, substitute the given number into the equation in place of \(x\).
• Then, perform the calculations with this substitution.

In our example, we substitute \(116\) for \(x\) and evaluate. This step is all about correctly inputting the value to see if it balances the equation on both sides.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to solve for unknowns. In this context, it involves

• performing mathematical operations such as multiplication and addition,
• and ensuring the order of operations is followed correctly.

For the equation \(-12x + 23 = -14\), after substituting \(x = 116\), we perform algebraic manipulations:1. Multiply: \(-12 \times 116\) calculates to \(-1392\).2. Add \(23\) to \(-1392\) resulting in \(-1369\).The goal is to simplify the left-hand side to directly compare with the right-hand side of the equation.

Verifying Solutions

Verifying a solution is an essential step in confirming whether a proposed value satisfies an equation. Once algebraic manipulation is performed, the efficacy of the solution must be checked.In the case of \(-12x + 23 = -14\), after simplifying the left-hand side to \(-1369\), we compare it with \(-14\).

• If both sides are equal, then the value \(x = 116\) is indeed a solution.
• However, since \(-1369eq -14\), the left and right sides are unequal, indicating that \(x = 116\) is not a solution.

This verification step is critical in ensuring that only true results are accepted when solving equations.