Problem 86 - Intermediate Algebra

Question

Solve each equation. $$ -(9 m-11.13)=7.7(6+m) $$

Step-by-Step Solution

Answer

The solution to the equation is \(m = -2.1\).

Step 1: Distribute the Negative Sign

We start by distributing the negative sign over the expression on the left side of the equation. The expression becomes \[-(9m - 11.13) = -9m + 11.13\].

Step 2: Expand the Right Side

Next, distribute the 7.7 on the right side of the equation:\[7.7(6 + m) = 7.7 imes 6 + 7.7 imes m = 46.2 + 7.7m\].

Step 3: Equate and Rearrange Terms

Now, plug the expanded expressions into the equation:\[-9m + 11.13 = 46.2 + 7.7m\]Rearrange to group like terms:\[-9m - 7.7m = 46.2 - 11.13\].

Step 4: Simplify the Equation

Combine like terms:\[-16.7m = 35.07\].

Step 5: Solve for m

Now, solve for \(m\) by dividing both sides by -16.7:\[m = \frac{35.07}{-16.7} = -2.1\].

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental algebraic principle. It allows you to multiply a single term by multiple terms inside a parenthesis. Essentially, it "distributes" multiplication over addition or subtraction.

In this exercise, the distributive property is applied twice: once to the negative sign on the left and once to the coefficient 7.7 on the right.

• On the left side: A negative sign is treated as multiplying by -1. It turned \(- (9m - 11.13)\) into \(-9m + 11.13\).
• On the right side: 7.7 multiplies both terms inside the parenthesis. Leading to \(7.7(6 + m) = 46.2 + 7.7m\).

Always remember that when you distribute, each term inside parentheses must be multiplied by the term outside.

Combining Like Terms in Equations

Combining like terms is a crucial skill when simplifying expressions. It involves adding or subtracting coefficients of terms with the same variable.

When given a should be simplified, terms that share the same variable part can be merged. Let’s see this in the exercise:

• The equation was \(-9m + 11.13 = 46.2 + 7.7m\).
• To simplify, all terms with \(m\) should be on one side. Leading to \(-9m - 7.7m = 46.2 - 11.13\).
• Which combines to make \(-16.7m = 35.07\).

This process reveals the simplest form of the equation. Making it much easier to see what the variable equals.

Solving for a Variable

Once your equation is simplified and like terms are combined, the next step is to solve for the variable.

This means isolating the variable on one side of the equation. For this you perform operations that simplify the equation step-by-step. For example:

• Our simplified equation from combining like terms was \(-16.7m = 35.07\).
• To isolate \(m\), divide both sides by \(-16.7\) to get \(m\) on its own.
• The division \(\frac{35.07}{-16.7}\) results in \(-2.1\).

This solution process ensures that you find the precise value of the unknown variable. Also remember: whatever operation you perform on one side must be done on the other. This keeps the equation balanced.