Problem 35 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ 8 x+2=5 x+1 ; x=-13 $$

Step-by-Step Solution

Answer

The given value \( x = -13 \) is not a solution to the equation.

Step 1: Substitute the given value

We need to check if the given value, \( x = -13 \), satisfies the equation \( 8x + 2 = 5x + 1 \). To do this, substitute \( -13 \) wherever there is an \( x \) in the equation.

Step 2: Simplify the left side

Substitute \( x = -13 \) in the left side of the equation: \( 8(-13) + 2 \). Calculate: \[ 8(-13) = -104 \] So, the left side becomes: \[ -104 + 2 = -102 \]

Step 3: Simplify the right side

Now, substitute \( x = -13 \) in the right side of the equation: \( 5(-13) + 1 \). Calculate: \[ 5(-13) = -65 \] So, the right side becomes: \[ -65 + 1 = -64 \]

Step 4: Compare both sides

We have found that the left side simplifies to \( -102 \) and the right side simplifies to \( -64 \). Since \( -102 eq -64 \), the given value \( x = -13 \) is not a solution to the equation \( 8x + 2 = 5x + 1 \).

Key Concepts

Solution Verification

In mathematics, solution verification is the process of determining whether a given value satisfies an equation. It involves substituting the proposed solution into the equation to see if it renders both sides equal. In our example, we are checking if the value \( x = -13 \) is a solution to the linear equation \( 8x + 2 = 5x + 1 \). By substituting \( x \) with \( -13 \), we aim to verify if the left side equals the right side. If they are not equal, then \( x = -13 \) is not a solution for this equation. This step is crucial in ensuring the validity of solutions, particularly when solving real-world problems where accuracy is important.

Substitution Method

The substitution method is a useful strategy in finding solutions to equations. It involves replacing a variable with a given or known value to see how it affects the equation. For our current exercise, replacing \( x \) with \( -13 \), we substitute this value into both sides of the original equation, \( 8x + 2 = 5x + 1 \). By performing this substitution:

• In the left side of the equation: Replace \( x \) with \( -13 \), resulting in \( 8(-13) + 2 \).
• In the right side of the equation: Also replace \( x \) with \( -13 \), leading to the expression \( 5(-13) + 1 \).

Substituting values is a fundamental algebraic technique that simplifies equations, aiding in verifying solutions or even deriving new ones. It's an approachable method that becomes especially useful when working with variables represented in more complex systems.

Equation Simplification

Equation simplification is the process of reducing an equation to its simplest form, making it easier to analyze and compare. Once a variable is substituted with a specific value, simplification involves calculating the numerical expressions to reduce the equation. In our case:

• Begin by simplifying the left side \( 8(-13) + 2 \):
  • Calculate \( 8 \times (-13) = -104 \)
  • Then, \( -104 + 2 = -102 \)
• Next, simplify the right side \( 5(-13) + 1 \):
  • Calculate \( 5 \times (-13) = -65 \)
  • Then, \( -65 + 1 = -64 \)

After simplification, we compare the results. The equation is not balanced as \( -102 eq -64 \). This shows the value \( x = -13 \) does not solve the equation. Simplification is a step that aids understanding by stripping an equation down to its basic components, making it easier to solve or check.