Problem 5 - Fundamentals of Mathematics

Question

Verify that -1 is a solution to \(6 m-5+2 m=7 m-6\).

Step-by-Step Solution

Answer

Yes, -1 is a solution, as both sides of the equation equal -13 when m = -1.

Step 1: Substitute the Value into the Equation

We start by substituting \( m = -1 \) into the equation \( 6m - 5 + 2m = 7m - 6 \). This gives us: \( 6(-1) - 5 + 2(-1) = 7(-1) - 6 \).

Step 2: Simplify Both Sides of the Equation

Simplify the left side: \( 6(-1) = -6 \) and \( 2(-1) = -2 \), so \( -6 - 5 - 2 = -13 \). Simplify the right side: \( 7(-1) = -7 \), so \( -7 - 6 = -13 \).

Step 3: Compare Simplified Expressions

Both sides of the equation equal \(-13\). Since the left side \(-13\) is equal to the right side \(-13\), the equation is true for \( m = -1 \). Therefore, \( m = -1 \) is indeed a solution to the equation.

Key Concepts

Linear equations

Linear equations are fundamental in algebra and describe relationships between variables using a straight line when graphed. They have one or more variables where each variable is raised to the first power. The general form of a linear equation in one variable is:

• In one variable: \( ax + b = 0 \)
• In two variables: \( ax + by = c \)

These equations are essential in mathematical modeling and practical problems like calculating distance, speed, costs, and profits. In the given exercise, the equation \( 6m - 5 + 2m = 7m - 6 \) is a linear equation in one variable \( m \). The task is to verify whether a specific value for \( m \), namely \(-1\), satisfies this equation.

Substitution method

The substitution method is a technique used to solve equations by replacing the variable with a specific value or expression. This approach is especially useful in verifying solutions. To apply the substitution method, follow these steps:

• Choose the value to substitute into the variable of the equation.
• Replace the variable with the chosen value.
• Simplify the equation to check if both sides are equal.

In our exercise, the substitution method is used by placing \( m = -1 \) into the equation \( 6m - 5 + 2m = 7m - 6 \), leading to calculations that verify whether \( m = -1 \) is indeed a solution.

Solving equations

Solving equations involves finding the value of the variable(s) that make the equation true. This requires manipulation such that one side of the equation reflects the variable's value directly or simplifies the equation enough to find the value indirectly.Key steps for solving linear equations include:

• Combining like terms to simplify the equation.
• Using algebraic operations (addition, subtraction, multiplication, and division) to isolate the variable.
• Ensuring the operation is performed equally on both sides to maintain balance.

In the provided solution, solving involves simplifying both sides by combining the terms associated with \( m \): \( 6m + 2m - 5 = 7m - 6 \). These steps ensure that the same value of \( m \) is calculated on both sides, confirming that they balance when \( m = -1 \).

Mathematical verification

Mathematical verification is the process of confirming that a chosen solution satisfies an equation. This is done by substituting the solution into the original equation and checking if both sides of the equation result in identical values. In the exercise, the verification process involves these steps:

• Substitute \( m = -1 \) into the equation \( 6m - 5 + 2m = 7m - 6 \).
• Simplify both sides: the left side simplifies to \(-13\) and the right side also simplifies to \(-13\).
• Compare both sides: both equal \(-13\), confirming the solution is correct.

Verification reassures that \( m = -1 \) is not simply calculated correctly but ensures the accuracy and correctness of the solution as per the equation's requirements.