Problem 89 - Elementary Algebra Within Reach

Question

Structure You could solve \(3(x-7)=15\) by applying the Distributive Property as the first step. However, there is another way to begin. What is it?

Step-by-Step Solution

Answer

The other way to begin solving the equation is by isolating the term with the variable, \(x-7\), first. This results in \(x = 12\).

Step 1: Isolate the term with the variable

Begin by isolating \(x-7\). This can be done by dividing each side of the equation by 3. This gives \((x-7) = \frac{15}{3}\).

Step 2: Simplifying the right side

Simplify the right side of the equation by performing the division. This gives \((x-7) = 5\).

Step 3: Solve for x

Finally, solve for x by adding 7 to each side of the equation. This yields \(x = 5 + 7\).

Key Concepts

Distributive Property

The Distributive Property is a fundamental principle in algebra that applies to multiplication over addition or subtraction. It might sound complex, but it can be simplified with a basic example. If you have an equation like \(a(b + c)\), this means you multiply \(a\) with both \(b\) and \(c\), resulting in \(ab + ac\). This property is particularly useful for breaking down equations, allowing you to simplify them by eliminating parentheses. In the exercise, the equation \(3(x-7)=15\) can be tackled using the Distributive Property by multiplying 3 across the terms inside the parenthesis:

• Apply: \(3 \times x = 3x\)
• Apply: \(3 \times -7 = -21\)

Thus, the equation becomes \(3x - 21 = 15\), which can be further simplified to solve for \(x\).

Isolating Variables

Isolating variables is a crucial step in solving equations where you need to find the value of unknown variables. The goal is to have the variable appear alone on one side of the equation. This helps in simplifying the process of solving. In our example, the first move to isolate the variable term \((x-7)\) is by dividing the entire equation by 3:

• Given: \(3(x-7)=15\)
• Divide by 3: \(x-7 = \frac{15}{3}\)

Now, the equation simplifies to \(x - 7 = 5\). Isolating the variable initially can save time and streamline the entire solving process.

Equation Simplification

To solve an equation efficiently, simplification is a necessary step. Simplifying an equation can involve several operations such as reducing fractions, adding, or subtracting terms. Once the variable term \(x - 7\) is isolated and the equation is rewritten as \(x - 7 = 5\), the next task is to simplify the right-hand side:

• Simplify: \(\frac{15}{3} = 5\)

With the simplified equation, where \(x - 7 = 5\), solving for \(x\) becomes a straightforward task by performing a simple addition.

Step-by-Step Problem Solving

Solving equations in a step-by-step manner breaks the process into manageable parts. This method allows for easy tracking of what needs to be done at each stage, and confirms that you are on track to find the right solution. In the given problem, after simplifying the initial equation to \(x - 7 = 5\), the final step involves solving for the variable \(x\):

• Add 7 to both sides: \(x - 7 + 7 = 5 + 7\)
• Simplify to get: \(x = 12\)

By approaching the problem in these small steps, each calculation is straightforward, making it less likely to make mistakes and ensuring that you fully understand each stage of the solving process.