Problem 41 - Algebra 1

Question

Solve the equation. $$-r-(-7)=16$$

Step-by-Step Solution

Answer

The solution to the equation \(-r-(-7)=16\) is \(r = -9\).

Step 1: Solving the equation Step 1: Simplify the equation

First we handle the parentheses and the negative sign in front it. The negative sign changes the signs of the terms inside the parentheses. So the equation becomes: \(-r + 7 = 16\).

Step 2: Solving the equation Step 2: Isolate the variable

To isolate the variable \(r\), subtract 7 from both sides of the equation. This forms the equation \(-r = 16 - 7\).

Step 3: Solving the equation Step 3: Simplify the right side

By subtraction, we have \(-r = 9\).

Step 4: Solving the equation Step 4: Solve for \(r\)

The variable \(r\) is still negative, to get it in the positive form, we multiply both sides by -1. Hence, we get: \(r = -9\).

Key Concepts

Simplifying Expressions

Simplifying an expression means to make it as neat as possible by performing the operations within it. Begin by removing any parentheses, as they often contain terms that can be simplified.
In the original exercise, there was a negative sign followed by a set of parentheses:

• The expression \(-r-(-7)=16\) was given.
• By distributing the negative sign, the equation was simplified to \(-r + 7 = 16\).

This step helps to make future operations easier, setting the stage for isolating the variable.

Isolating Variables

Isolating the variable is the process of rearranging an equation so that the unknown value is by itself on one side of the equation. It's like solving a puzzle one piece at a time.
In the exercise, the goal was to have the variable \(r\) alone on one side.

• Start by moving constants to the opposite side of the equation. In this case, subtract 7 from both sides to get: \-r = 16 - 7\.
• After the subtraction, the equation simplifies to \-r = 9\.

This step is crucial in solving equations, as it leads you closer to finding the exact value of the unknown variable.

Properties of Operations

When you solve equations, you use these properties to rearrange and simplify expressions. Let's look at some key properties that were applied in the solution:

• Additive Inverse Property: This was used when \(-(-7)\) was simplified to \(+7\). The additive inverse of a number is its opposite.
• Subtraction: From both sides, \(7\) was subtracted to keep the equation balanced. Whatever you do to one side, you must do to the other.
• Multiplicative Inverse Property: Finally, by multiplying both sides by \(-1\) to solve for \(-r = 9\), the variable was isolated in a positive form \(r = -9\).

Using these properties systematically helps maintain equality while simplifying the equation.

Integer Operations

Understanding integer operations is crucial for solving equations effectively. Integers include negative numbers, positive numbers, and zero. Here, specific integer operations were used:

• Absolute Value Changes: Manipulating the negative sign around integers, such as when transitioning from \(-(-7)\) to \(+7\), uses knowledge of absolute values.
• Subtraction: When handling \-r = 9\, subtraction removes constants to simplify potential integer values. \(16 - 7\) equated to \(9\).
• Multiplication of Integers: Lastly, when solving for \(r\), multiplying by \(-1\) changed the sign of \(-r\) to get \(r = -9\).

Mastering these operations allows you to efficiently solve for unknown variables in any given equation.