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Chapter 2: Problem 45

Solve each equation. $$-(j+2)+2 j-1=5$$

Short Answer

Expert verified
j = 8

Step by step solution

01

- Distribute the negative sign

First, distribute the negative sign through the parentheses: -(j+2) becomes -j - 2 The equation now looks like this: - j - 2 + 2 j - 1 = 5
02

- Combine like terms

Combine the 'j' terms and the constants: - j + 2 j = j - 2 - 1 = -3 The equation now simplifies to: j - 3 = 5
03

- Solve for j

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
When solving linear equations, the distributive property allows you to simplify expressions by distributing a constant or variable across terms within parentheses. Let's start with the equation from the example: \( -(j + 2) + 2j - 1 = 5 \). Here, you have a negative sign outside the parentheses. You're distributing this negative sign to each term inside the parentheses: \( -1 \times (j) = -j \) and \( -1 \times (2) = -2 \). The equation then transitions to: \( -j - 2 + 2j - 1 = 5 \). By distributing the negative sign, you break down complex expressions, making the equation more manageable.
Combining Like Terms
Combining like terms is a crucial step in simplifying a linear equation. 'Like terms' are terms that have the same variables raised to the same power. In the equation \( -j - 2 + 2j - 1 = 5 \), you can combine \( -j \) and \( 2j \) because they both have the variable 'j': \( -j + 2j = j \). You can also combine the constants \( -2 \) and \( -1 \): \( -2 - 1 = -3 \). After combining like terms, the equation simplifies to \( j - 3 = 5 \). This step consolidates terms, reducing the equation to a more straightforward form.
Isolating Variables
The final step in solving a linear equation is isolating the variable. This means getting the variable on one side of the equation and the constants on the other side. Starting from the simplified equation \( j - 3 = 5 \), you need to isolate \( j \). To do this, add 3 to both sides of the equation. This keeps the equation balanced: \( j - 3 + 3 = 5 + 3 \). Simplifying both sides gives you: \( j = 8 \). Isolating the variable allows you to find the solution to the equation. In this case, \( j \) equals 8. By following these steps鈥攄istributing, combining like terms, and isolating the variable鈥攜ou can systematically solve any linear equation.

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