Problem 6 - Algebra and Trigonometry

Question

Determine whether the given value is a solution of the equation. \(2-5 x=8+x\) (a) \(x=-1 \quad\) (b) \(x=1\)

Step-by-Step Solution

Answer

(a) Yes, (b) No.

Step 1: Substitute x = -1 into the equation

First, we need to check if \(x = -1\) is a solution of the equation \(2 - 5x = 8 + x\). Substitute \(-1\) for \(x\) in the equation: \[2 - 5(-1) = 8 + (-1)\] Calculate each side separately. For the left side:\[2 + 5 = 7\]For the right side:\[8 - 1 = 7\] Since both sides are equal, \(x = -1\) is a solution.

Step 2: Substitute x = 1 into the equation

Now, check if \(x = 1\) is a solution of the equation \(2 - 5x = 8 + x\). Substitute \(1\) for \(x\) in the equation:\[2 - 5(1) = 8 + 1\]Calculate each side separately. For the left side:\[2 - 5 = -3\]And for the right side:\[8 + 1 = 9\]Since both sides are not equal, \(x = 1\) is not a solution.

Key Concepts

Solution Verification

Solution verification is a simple yet vital process in solving linear equations. It involves substituting a given value into the equation to see if it satisfies the equation, meaning if both sides of the equation equal each other after the substitution.

For example, in our exercise, we start with the equation \(2-5x=8+x\). We are given two potential values for \(x\), specifically \(x=-1\) and \(x=1\), which we test one by one. To verify if \(x=-1\) is a solution, we substitute \(-1\) for \(x\):

• Left side: \(2 - 5(-1) = 2 + 5 = 7\)
• Right side: \(8 + (-1) = 8 - 1 = 7\)

When both sides equal \(7\), it confirms \(x=-1\) is indeed a solution.

This method helps in ensuring the correctness of potential solutions, making it an essential part of solving equations.

Equation Solving

Equation solving is the process of finding one or more values which satisfy the given equation. In the realm of linear equations, this often involves manipulating the equation to isolate the variable of interest on one side of the equation.

For our equation \(2-5x=8+x\), to solve for \(x\), we need to investigate whether any potential solutions make both sides equal after substitution. As demonstrated, when substituting \(x = -1\), we observed equal values on both sides, confirming it as a solution. Conversely, for \(x = 1\):

• Left side: \(2 - 5(1) = 2 - 5 = -3\)
• Right side: \(8 + 1 = 9\)

The inequality \(-3 eq 9\) signifies that \(x = 1\) does not solve the equation.

This step-by-step approach can simplify understanding how solutions to linear equations are found.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them more manageable. This technique is often applied in solving equations to isolate a variable or simplify problems.

When dealing with the equation \(2-5x=8+x\), algebraic manipulation helps us see potential solutions by working through the equation logically. Suppose we want to manipulate the equation directly before substituting values. The goal is to isolate \(x\) by:

• Subtracting \(x\) from both sides to get \(2 - 6x = 8\)
• Subtracting 2 from both sides, resulting in \(-6x = 6\)
• Dividing each side by \(-6\) gives \(x = -1\)

These steps reveal that manipulating the equation directly would have revealed \(x=-1\) as a solution.

This process shows the power of algebra in breaking down complex problems into simpler steps.