Problem 12 - Algebra 1

Question

Check whether the given number is a solution of the equation. $$6 x-4(9-x)=106 ; 7$$

Step-by-Step Solution

Answer

No, the number 7 is not a solution to the equation \(6x - 4(9 - x) = 106\).

Step 1 - Substitute

Replace x with the given number \(7\) in the equation, resulting in \(6(7) - 4(9 - 7) = 106\).

Step 2 - Simplify

Simplify the equation by first dealing with the parenthesis \(9 - 7 = 2\). Now rewrite the equation with the new value inside the parenthesis, \(6(7) - 4(2) = 106\). Again simplify the equation by multiplying \(6*7 = 42\) and \(4*2 = 8\), so it becomes \(42 - 8 = 106\).

Step 3 - Final Evaluation

Evaluate the expression on the left side of the equation \(42 - 8 = 34\). Now, check if it matches with the given number on the right side of the equation, which is 106. In this case, they are not equal.

Key Concepts

Substitution Method

The substitution method is a technique that helps us solve equations by replacing variables with specific values. In the context of our equation, we have to substitute the variable \(x\) with the number \(7\). This is how we check if \(7\) is actually a solution to the equation.
First, take the original equation: \[ 6x - 4(9 - x) = 106 \]Replace \(x\) with \(7\). Now, the equation looks like this: \[ 6(7) - 4(9 - 7) = 106 \]
This transformation allows us to work only with numbers, confirming whether or not they satisfy the equation. This step is crucial since it lays the groundwork for further simplification.

Simplification

Simplification is key in solving any equation effectively. It involves restructuring the equation to make it easier to understand and solve. In our exercise, the simplification process started immediately after the substitution.
First, handle any arithmetic operations. Evaluate the expression inside the parentheses: \[ 9 - 7 = 2 \]
This leads to a simpler expression: \[ 6(7) - 4(2) = 106 \]
Then carry out multiplications:

• \( 6 \times 7 = 42 \)
• \( 4 \times 2 = 8 \)

Once we've simplified the terms, the equation evolves into: \[ 42 - 8 = 106 \]
Through simplification, calculations become straightforward, making the next steps more accessible.

Parentheses Evaluation

Evaluating expressions within parentheses is an important part of simplifying equations. Perform these calculations first, following the order of operations (PEMDAS/BODMAS).
Here, the parentheses \( (9 - x) \) required evaluation. After substituting \(x\) with \(7\), we calculated: \[ 9 - 7 = 2 \]
This step prepares us to rewrite the equation without the parentheses, turning complex expressions into simpler, number-focused tasks reliant on basic arithmetic.

Verification of Solutions

Verification of solutions is the final step to ensure that our calculations are correct. After simplifying the left side of the equation, we aim to see if it matches the right side.
Our calculations gave us: \[ 42 - 8 = 34 \]
Unfortunately, this does not equal the right side of the original equation, \( 106 \). This tells us that replacing \(x\) with \(7\) does not solve the equation.
Verification is essential, as it confirms whether the proposed solution is valid. If it doesn't work, adjustments or reconsiderations may be needed to find the correct solution.