Problem 10 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ -8 x-33=3 x ; \quad x=3 $$

Step-by-Step Solution

Answer

No, \( x = 3 \) is not a solution because \( -57 \neq 9 \).

Step 1: Substitute x in the Equation

First, we take the given value, \( x = 3 \), and substitute it into the equation \( -8x - 33 = 3x \). This gives us: \( -8(3) - 33 = 3(3) \).

Step 2: Simplify Both Sides of the Equation

Next, simplify both sides of the equation. On the left side, calculate \( -8 \times 3 \) which is \( -24 \), so \( -24 - 33 \). On the right side, calculate \( 3 \times 3 \) which is \( 9 \). Now the equation is \( -24 - 33 = 9 \).

Step 3: Simplify Further

Simplify the left side of the equation further. Calculate \( -24 - 33 \), which equals \( -57 \). The equation now is \( -57 = 9 \).

Step 4: Compare Both Sides of the Equation

Now compare the simplified left side \( -57 \) with the right side \( 9 \). Since \( -57 eq 9 \), the given value \( x = 3 \) is not a solution to the equation.

Key Concepts

Solution Verification

When solving a linear equation, an essential step is to verify whether a proposed value is indeed a solution. Solution verification involves checking if both sides of the equation hold equal after substituting the value. This ensures the correctness of the solution.

To verify if the value \( x = 3 \) is a solution for the equation \( -8x - 33 = 3x \), we replace \( x \) with 3 in the equation.

• Substitute \( x = 3 \) into the equation: \( -8(3) - 33 = 3(3) \).
• Solve both sides to check if they equate.

After simplification, if both sides do not match, then \( x = 3 \) is not a solution. In our original exercise, the left side simplified to \( -57 \), whereas the right was \( 9 \), showing that \( x = 3 \) is not a solution.

Substitution Method

The substitution method is a powerful tool in solving equations. It involves replacing variables in an equation with given or previously determined values. This technique is very useful when testing potential solutions because it allows one to directly evaluate the equation without rearranging or balancing it.

In our exercise, we used the substitution method to test \( x = 3 \). Here’s how it works:

• Take the equation: \( -8x - 33 = 3x \).
• Substitute \( x = 3 \) into the equation to get: \( -8(3) - 33 = 3(3) \).

By doing so, you can directly calculate the result of both sides of the equation with the given value. This helps in checking the correctness efficiently.

Simplifying Equations

Simplifying equations is a crucial aspect of algebra that involves reducing an expression to its simplest form. This step makes it easier to compare different terms and helps in identifying solutions.

In our example, simplifying played a key role in verifying the solution:

• Start by calculating each term after substitution: \( -8(3) = -24 \) and \( 3(3) = 9 \).
• Then simplify terms to reduce complexity: combine \( -24 - 33 \) on the left side to get \( -57 \).

By simplifying, you can see clearly that the two sides of the equation are not equal (left side: \(-57\), right side: \(9\)). This direct comparison shows that \( x = 3 \) does not satisfy the equation, helping us understand the nature of the solution better.