Problem 5 - Algebra and Trigonometry

Question

Determine whether the given value is a solution of the equation. \(4 x+7=9 x-3\) (a) \(x=-2 \quad\) (b) \(x=2\)

Step-by-Step Solution

Answer

(a) No, (b) Yes.

Step 1: Substitute the value into the equation for option (a)

We have the equation \(4x + 7 = 9x - 3\). Substitute \(x = -2\) into the equation: \(4(-2) + 7 = 9(-2) - 3\).

Step 2: Simplify both sides for option (a)

Calculate the left side: \(4(-2) + 7 = -8 + 7 = -1\). Calculate the right side: \(9(-2) - 3 = -18 - 3 = -21\).

Step 3: Compare the sides for option (a)

The left side is \(-1\) and the right side is \(-21\). Since \(-1 eq -21\), \(x = -2\) is not a solution to the equation.

Step 4: Substitute the value into the equation for option (b)

Substitute \(x = 2\) into the equation: \(4(2) + 7 = 9(2) - 3\).

Step 5: Simplify both sides for option (b)

Calculate the left side: \(4(2) + 7 = 8 + 7 = 15\). Calculate the right side: \(9(2) - 3 = 18 - 3 = 15\).

Step 6: Compare the sides for option (b)

The left side is \(15\) and the right side is \(15\). Since both sides are equal, \(x = 2\) is a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a popular technique used to solve equations, particularly linear equations. It involves replacing a variable with a given value in the equation to find out if that value satisfies the statement. This is a straightforward process that requires careful attention to detail. Here's how you do it:

• Take the equation and identify the variable for which you want to check the solution.
• Substitute this variable with the given value into the equation.
• Solve the resulting equation to see if both sides equal one another.

By substituting the values one by one, as shown in the original exercise, calculations are performed directly. This method is not just limited to one value; you can check multiple values to find the correct solution or solutions.

Linear Equations

Linear equations are mathematical statements that express a relationship between variables. They are generally written in the form of ax + b = cx + d, where x represents the variable, and a, b, c, and d are constants. In the equation from the original exercise, we have coefficients and constants interacting with the variable x. Characteristics of linear equations include:

• They form a straight line when graphed on a coordinate plane.
• They have exactly one solution unless both sides simplify to an identity (resulting in infinite solutions) or a contradiction (resulting in no solution).

Understanding linear equations is crucial, as they form the basis of algebra. Their simplicity and straightforwardness make them a great starting point for solving more complex equations later on.

Solution Verification

Solution verification is the process of confirming whether a given value is a solution to an equation. This ensures the accuracy of transactions, especially during substitutions in equations. In the original exercise, solution verification occurs with these steps:

• Compare the two sides of the equation after substitution and simplification.
• If both sides are equal, the substituted value is indeed a solution.
• If the sides do not align, it means the value is not a solution.

This process is essential for validating our work in algebra. It enables us to guarantee that our found solutions satisfy the initial conditions set by the equation. Solution verification helps in affirming the correctness of results, aiding in the development of confidence in mathematical problem-solving.