Problem 57 - Elementary Algebra

Question

Solve. $$ 3(x+5)-2(2 x+3)=7 x+9 $$

Step-by-Step Solution

Answer

The solution for x is 0.

Step 1: Expand the Expressions

Begin by expanding the terms inside the parentheses on both sides of the equation. For the left side, distribute 3 into \( (x+5) \) and -2 into \( (2x+3) \). The expression becomes:\[3 \times x + 3 \times 5 - 2 \times 2x - 2 \times 3\]This simplifies to:\[3x + 15 - 4x - 6\]

Step 2: Simplify the Left Side

Now, simplify the expression by combining like terms on the left side. Combine \(3x\) and \(-4x\):\[-x\]Combine \(15\) and \(-6\):\[+ 9\]So, the left side simplifies to:\[-x + 9\]

Step 3: Write the New Equation

Rewrite the original equation with the simplified expressions:\[-x + 9 = 7x + 9\]

Step 4: Move Variables to One Side

Move all the x terms to one side by adding \(x\) to both sides of the equation:\[-x + x + 9 = 7x + x + 9\]This simplifies to:\[9 = 8x + 9\]

Step 5: Isolate the Variable Term

Subtract 9 from both sides to isolate the term containing x:\[9 - 9 = 8x + 9 - 9\]This gives:\[0 = 8x\]

Step 6: Solve for x

To solve for x, divide both sides of the equation by 8:\[\frac{0}{8} = \frac{8x}{8}\]This gives:\[x = 0\]

Key Concepts

Expand Expressions

In the context of solving linear equations, expanding expressions is an important first step. It involves distributing the multiplication across terms inside parentheses to simplify the equation. For example, in the equation \(3(x+5)-2(2x+3)\), we expand by multiplying 3 by each term inside the first set of parentheses \((x+5)\), resulting in \(3x+15\). Next, multiply -2 across \((2x+3)\), giving \(-4x-6\). This process helps to eliminate parentheses and simplifies the equation to:

• \(3x + 15\) from \(3(x+5)\)
• \(-4x - 6\) from \(-2(2x+3)\)
• Combine to make: \(3x + 15 - 4x - 6\)

Expanding expressions correctly is crucial for making further steps easier, as it leads to a simplified and manageable equation.

Combining Like Terms

Once expressions are expanded, the next step is to combine like terms. 'Like terms' are terms that have identical variable parts. In our example, the expanded expression becomes \(3x + 15 - 4x - 6\). Here, we look for terms that we can combine:

• \(3x\) and \(-4x\) are like terms because they both contain the variable \(x\).
• Combining these gives \(-x\).
• The constants \(15\) and \(-6\) are also like terms.
• Combining these results in \(+9\).

In this step, the expression simplifies to \(-x+9\). Combining like terms reduces the equation's complexity, making it easier to isolate variables in later steps.

Isolating Variables

Isolating the variable means getting the variables on one side of the equation, which makes solving for the unknown easier. In the equation \(-x + 9 = 7x + 9\), we start by moving all terms containing the variable \(x\) to one side. To do this:

• Add \(x\) to both sides to eliminate \(-x\) on the left: \(-x + x + 9 = 7x + x + 9\).
• After simplifying, we get \(9 = 8x + 9\).
• Subtract \(9\) from both sides which gives us \(0 = 8x\).

This action concentrates all \(x\) terms on one side and constants on the other, setting up the equation for the final solution step. Isolating variables is vital for solving any equation because it allows us to focus on finding the value of the unknown.

Algebraic Manipulation

Algebraic manipulation is the process of using mathematical operations to solve for variables. It involves adjusting and rearranging the equation to isolate the unknown. In the last step, after isolating variables, we have \(0 = 8x\). To solve for \(x\), we use algebraic manipulation as follows:

• Divide both sides by 8: \(\frac{0}{8} = \frac{8x}{8}\).
• This results in \(x = 0\).

Through algebraic manipulation, we apply operations like addition, subtraction, multiplication, and division to both sides. This maintains equality and systematically breaks down the equation to find the value of the unknown. It is a fundamental part of solving linear equations, as it translates the rearranged equation into an answer.