Problem 66 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 6 a solution of \(2 x+7=3 x ?\)

Step-by-Step Solution

Answer

No, 6 is not a solution.

Step 1: Substitute the Number

To determine if 6 is a solution of the equation \(2x + 7 = 3x\), substitute \(x\) with 6 in the equation. This gives us: \(2(6) + 7\) on the left-hand side.

Step 2: Simplify the Left-Hand Side

Calculate \(2(6)\) which equals 12, then add 7. This simplifies the left-hand side to 19. So the left side is \(19 = 3x\).

Step 3: Simplify the Right-Hand Side

Substitute \(x = 6\) on the right side of the equation to get \(3(6)\). Calculating this gives us 18.

Step 4: Compare Both Sides

Now compare the simplified left side (19) with the right side (18). Since 19 is not equal to 18, the original equation \(2x + 7 = 3x\) is not true when \(x = 6\).

Key Concepts

Solution of Equation

When determining if a specific value is a solution to an equation, our goal is to check whether the equation holds true when this value replaces the variable.
In mathematical terms, an equation is a statement that shows the equality between two expressions. For example, in the equation \(2x + 7 = 3x\), we are determining if 6 can be the value for \(x\) to keep both sides equal.
This involves a series of mathematical steps, which include substituting the variable and comparing both sides of the equation. Ensuring both expressions on either side are equal after substitution confirms that the chosen value is indeed a solution.
Conversely, if they are not equal, the chosen number is not a solution.

Substitution

Substitution is a fundamental technique in algebra that involves replacing a variable with a given numerical value.
This makes it possible to evaluate an equation or expression. In our case, we substitute \(x\) with 6 in the equation \(2x + 7 = 3x\).
This substitution transforms the equation and allows us to perform numerical calculations with real numbers rather than variables.

• Transformed left-hand side: \(2(6) + 7\)
• Transformed right-hand side: \(3(6)\)

Substituting correctly is critical for obtaining the right results. Misplacing the substituted value can lead to incorrect conclusions.

Simplifying Equations

Simplifying equations involves carrying out all arithmetic operations to rewrite the expressions in their simplest form.
This often means performing operations like addition, subtraction, multiplication, or division until you cannot simplify further. For the given equation:

• After substituting \(x = 6\), the left side becomes: \(2 \times 6 + 7 = 12 + 7 = 19\)
• The right side becomes: \(3 \times 6 = 18\)

These steps help in reducing each side of the equation to a single numerical value, which makes the comparison straightforward.
It’s an essential skill in algebra as it simplifies complex expressions and aids in solving equations efficiently.

Comparison of Expressions

Once an equation has been simplified on both sides, comparing the resultant numerical values is crucial. Comparing these values determines whether the statement of equality in the equation holds true.
For our exercise:

• Left side of the equation after substitution: 19
• Right side of the equation after substitution: 18

Recognizing that these numbers are not equal tells us that 6 is not a solution to the equation \(2x + 7 = 3x\).
Understanding this process is key to problem-solving in algebra, as it allows students to verify possible solutions or reject them based on their outcomes.