Problem 24 - Algebra 1

Question

Is the number given a solution of the equation? $$3 x-4=12-5 x ; 2$$

Step-by-Step Solution

Answer

Yes, 2 is a solution of the given equation.

Step 1: Identifying the Problem

The equation is given as \(3x - 4 = 12 - 5x\). It is required to find whether \(x = 2\) is a solution of the equation or not.

Step 2: Substituting Value

Substitute \(x = 2\) into the equation \(3x-4=12-5x\). So, the equation becomes \(3*2 - 4=?12 - 5*2\).

Step 3: Simplify the Equation

Do the arithmetic to simplify both sides of the equation. \(3 * 2 - 4 = 6 - 4 = 2\), and for the other side of the equation, \(12 - 5 * 2 = 12 - 10 = 2\).

Step 4: Verifying Equality

After the simplification, both sides of the equation are equal (2 = 2). Therefore, \(x = 2\) is indeed a solution of the equation.

Key Concepts

Substitution

Substitution is a primary method for checking solutions in linear equations. When you suspect that a number is a solution to an equation, you "substitute" this number in place of the variable. In the provided exercise, we want to examine if \(x = 2\) satisfies the equation \(3x - 4 = 12 - 5x\). This means we replace every instance of \(x\) with \(2\) and see if both sides of the equation balance out.
The term substitution might sound complex, but it's straightforward:

• Replace each occurrence of \(x\) with the number 2.
• Rewrite the equation as \(3 \cdot 2 - 4 =? 12 - 5 \cdot 2\).
• This simplifies the problem and sets the stage for evaluating the expression.

By carrying out this substitution, you essentially transform an algebraic problem into a simple arithmetic one, paving the way for easier evaluation of the solution's validity.

Arithmetic Simplification

Once you've substituted the value into the equation, the next step is to simplify the expressions on both sides of the equation. Arithmetic simplification involves performing basic computations like addition, subtraction, multiplication, or division to condense the equation.
In our example, after substituting \(x = 2\), we simplify both sides separately:

• For the left side: calculate \(3 \times 2 - 4\), which turns into \(6 - 4\) and results in \(2\).
• For the right side: compute \(12 - 5 \times 2\), which becomes \(12 - 10\) and also results in \(2\).

The primary goal here is to simplify each side to a single number or expression, making comparison easier in the verification step. This simplification involves straightforward arithmetic, reinforcing the concept that the end solution must satisfy both sides of the equation.

Verification of Solutions

After simplifying both sides of the equation, the task now is to compare the results to verify if the tested value is a true solution. Verification of solutions is crucial in proving that a specific number solves the equation.
For the equation \(3x - 4 = 12 - 5x\) with \(x = 2\):

• Both sides simplified to \(2\) after substitution and arithmetic simplification.
• Since \(2 = 2\) holds true, it confirms that both expressions are equal when \(x = 2\).

This verification not only establishes \(x = 2\) as a solution but also strengthens our understanding of solutions in algebra, emphasizing that both sides must reach the same value for a number to be considered a solution. This process builds a foundation for tackling more complex equations that employ the same principles.