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Chapter 8: Problem 36

In the following exercises, solve each equation. $$ -(j+2)+2 j-1=5 $$

Short Answer

Expert verified
j = 8

Step by step solution

01

Distribute and Simplify the Equation

First, distribute the negative sign through the parentheses: \(-1(j+2) + 2j - 1 = 5\). This simplifies to \(-j - 2 + 2j - 1 = 5\).
02

Combine Like Terms

Next, combine the like terms on the left side of the equation: \(-j + 2j - 2 - 1 = 5\). This simplifies to \(j - 3 = 5\).
03

Isolate the Variable

Add 3 to both sides of the equation to isolate \(j\): \(j - 3 + 3 = 5 + 3\). This simplifies to \(j = 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is essential in solving linear equations. It allows you to eliminate parentheses by distributing the factor outside the parentheses to each term inside. In the exercise, we have an equation with a negative sign before the parentheses: \(-1(j+2) + 2j - 1 = 5\). To apply the distributive property, we multiply \(-1\) by each term inside the parentheses: \(-1 \cdot j + (-1) \cdot 2 = -j - 2\). This simplifies the equation to \(-j - 2 + 2j - 1 = 5\). Understanding and correctly applying the distributive property is the first step to solve more complex equations.
Combining Like Terms
Combining like terms means consolidating terms that have the same variable part. This simplification helps in reducing the equation to an easier form. In our example, we have \(-j - 2 + 2j - 1 = 5\). Here, \(-j\) and \(+2j\) are like terms since both involve the variable \j\. When we combine them, we get \(-j + 2j = j\). Hence, the equation simplifies to \(j - 3 = 5\) when \-2\ and \-1\ are also combined. Always remember to combine similar variable terms and constants separately.
Isolating Variables
Isolating the variable means manipulating the equation to get the variable on one side. This is the final step in solving for the variable. In the example, after simplifying to \(j - 3 = 5\), we need to isolate \j\. To do this, we add \3\ to both sides of the equation: \(j - 3 + 3 = 5 + 3\). This gives us \(j = 8\). Now the variable is isolated, and we have our solution. Isolating the variable often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number to maintain equality.

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Most popular questions from this chapter

In the following exercises, translate to an equation and then solve. The sum of \(-9 m\) and 10\(m\) is \(-25\)

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In the following exercises, translate to an equation and then solve. The sum of \(y\) and \(-3\) is 40

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In the following exercises, solve each linear equation using the general strategy. $$ 8(r-2)=6(r+10) $$
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