Problem 30 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$5 x-7=-7+2(x+3)$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Simplify Both Sides

First, simplify both sides of the equation. Start by distributing the 2 on the right-hand side:\[ 5x - 7 = -7 + 2(x + 3) \]which simplifies to:\[ 5x - 7 = -7 + 2x + 6 \]Now simplify the right side by combining like terms:\[ 5x - 7 = 2x - 1 \]

Step 2: Move all Terms Involving x to One Side

To isolate the variable terms on one side, subtract \(2x\) from both sides:\[ 5x - 2x - 7 = 2x - 2x - 1 \]which simplifies to:\[ 3x - 7 = -1 \]

Step 3: Isolate the x-Term

Add 7 to both sides to move the constant term:\[ 3x - 7 + 7 = -1 + 7 \]which simplifies to:\[ 3x = 6 \]

Step 4: Solve for x

Divide both sides by 3 to solve for \(x\):\[ \frac{3x}{3} = \frac{6}{3} \]which gives us:\[ x = 2 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions and solve equations easily. When you distribute a number across terms within parentheses, it means multiplying the number by each term inside. In our original exercise, we have the equation: \[ 5x - 7 = -7 + 2(x + 3) \]. Here, you need to distribute the 2 across both the terms inside the parenthesis. This means:\[ 2(x + 3) = 2 \cdot x + 2 \cdot 3 \].
This results in: \[ 2x + 6 \]. Now you replace the original part of the equation on the right-hand side so that it reads: \[ -7 + 2x + 6 \].
Using the distributive property makes managing more complicated looking equations far simpler, allowing for success with further simplification.

Combining Like Terms

Combining like terms is an essential skill when solving equations. This means grouping together terms that have the same variable raised to the same power. For example, in the expression \[ 2x + 2 + 3x + 4 \], you can combine the terms with \(x\) together and the constant terms separately.
In the exercise example, after using the distributive property, the equation becomes: \[ 5x - 7 = -7 + 2x + 6 \].

• You can see there are like terms with \(x\) on both sides: \(5x\) on the left and \(2x\) on the right.
• Additionally, the constants on the right, \(-7\) and \(+6\), can be combined.

Combining them gives us: \[ 5x - 7 = 2x - 1 \].
This crucial step simplifies the work needed to isolate the variable \(x\) effectively.

Isolating Variables

The ultimate goal in solving an equation is to isolate the variable, typically represented here by \(x\). Isolating a variable means getting it by itself on one side of the equation.
In this exercise, once the terms have been distributed and like terms combined, you're left with: \[ 5x - 7 = 2x - 1 \].

• You first move \(2x\) from the right side by subtracting \(2x\) from both sides, leading to: \[ 3x - 7 = -1 \].
• Next, to remove \(-7\), you add \(7\) to each side, simplifying to: \[ 3x = 6 \].

Finally, you isolate \(x\) by dividing both sides by \(3\). This gives you \(x = 2\).
Isolating variables allows you to determine their value, ultimately solving the equation.