Problem 8 - Precalculus

Question

Solve each equation. $$2 m-1+3 m+5=6 m-8$$

Step-by-Step Solution

Answer

The solution to the equation is \( m = 12 \).

Step 1: Combine Like Terms

The first step is to combine like terms on the left side of the equation. The like terms are those with the variable \( m \). Add \( 2m \) and \( 3m \) together to get \( 5m \). The equation becomes \( 5m - 1 + 5 = 6m - 8 \).

Step 2: Simplify Constant Terms

Next, combine the constant terms on the left side: \(-1 + 5 = 4\). So now, the equation is \( 5m + 4 = 6m - 8 \).

Step 3: Isolate the Variable Term

Subtract \( 5m \) from both sides to get the variable terms on one side: \( 4 = m - 8 \).

Step 4: Solve for the Variable

Add 8 to both sides to solve for \( m \): \( 4 + 8 = m \), which simplifies to \( m = 12 \).

Key Concepts

Understanding Like Terms

Like terms are essential when solving equations. They allow us to simplify the equation by combining similar elements. In our exercise, the like terms are those with the variable \( m \).

• "Like terms" refers to terms that have the same variable raised to the same power. For instance, \( 2m \) and \( 3m \) are like terms because they both include the same variable \( m \).

Identifying like terms makes it easier to condense an equation. This process involves simply adding or subtracting the coefficients of the like terms. Consider the example \( 2m + 3m \), which simplifies to \( 5m \). Recognizing and combining like terms simplifies the equation, leaving us with fewer terms to work with moving forward.

Methods of Variable Isolation

Variable isolation is a critical step in solving equations. It involves arranging the equation so the variable you're solving for appears on one side of the equation all by itself. Once like terms are combined, it helps to remove any coefficients or constants surrounding the variable.

• In our example, after combining the like terms, we subtract \( 5m \) from both sides. This step helps shift all terms containing \( m \) to one side of the equation, chiseling down the equation further to \( 4 = m - 8 \).

Variable isolation is about making the equation simpler and more direct. The ultimate goal is to solve for the variable in question. Doing so requires consistent inverse operations (like subtraction to counter addition). Breaking the problem into these steps will ultimately guide you to the solution of the equation.

Steps to Simplifying Equations

Simplifying equations is about making them as straightforward as possible, so solving them becomes easier. It starts with distributing multiplication over addition and using basic arithmetic to deal with like terms and constant terms.

• After isolating the variable in the example equation, we need to manage the number terms by adding or subtracting them accordingly. Simplifying happens when, for example, we take \(-1 + 5 = 4\) on the left side of the equation, reducing the equation further to \( 5m + 4 \).

Simplifying involves working through each segment of the equation methodically until each piece is as reduced as it can be. Paying attention to details and ensuring precision in each operation is critical. By carefully breaking down each step, you make the solving process faster and less prone to mistakes.