Problem 34 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(x+4(x+3)=17\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Distribute and Simplify

First, distribute the 4 to both the terms inside the parenthesis. This will result in: \(x + 4x + 12 = 17\). Then, simplify the equation by combining like terms: \(5x + 12 = 17\)

Step 2: Isolate x

Next, isolate \(x\) by subtracting 12 from both sides of the equation: \(5x = 5\)

Step 3: Solve for x

Finally, solve for \(x\) by dividing both sides of the equation by 5: \(x = 1\)

Key Concepts

Distributive Property

The distributive property is a useful tool when solving equations with parentheses. It helps to simplify the expression by eliminating the parentheses and spreading the multiplication over the terms inside.

• This principle can be mathematically represented as: \(a(b + c) = ab + ac\).
• In the exercise \(x + 4(x + 3) = 17\), you need to distribute the 4 across both \(x\) and the 3 inside the parentheses.
• So, multiply 4 with \(x\) to get \(4x\) and 4 with 3 to get 12. Therefore, the expression changes to \(x + 4x + 12 = 17\).

By expanding the equation with the distributive property, you've taken a step towards simplifying and solving the equation effectively.

Combining Like Terms

Once we break down the equation using the distributive property, we often end up with similar terms that can be combined. Combining like terms helps to further simplify and structure the equation, making it easier to solve.

• "Like terms" are terms that contain the same variable raised to the same power. For example, \(x\) and \(4x\) are like terms, but \(x\) and \(x^2\) are not.
• In the equation \(x + 4x + 12 = 17\), the like terms \(x\) and \(4x\) can be combined by adding the coefficients together, resulting in \(5x\). This step simplifies the equation to \(5x + 12 = 17\).

Through combining like terms, we reduce the complexity of the equation, allowing us to move closer to isolating the variable.

Isolating Variables

After simplifying the equation by combining like terms, the next goal is to isolate the variable on one side of the equation. This means getting the variable \(x\) by itself.

• To isolate \(x\) in the equation \(5x + 12 = 17\), begin by removing the constant term from the left side. You do this by subtracting 12 from both sides of the equation. This leaves us with \(5x = 5\).
• The next step is to solve for \(x\) by dividing both sides of the equation by 5, which gives \(x = 1\).

Isolating the variable is crucial because it ultimately reveals the solution to the equation. By properly handling the operations, we find that the solution is \(x = 1\), completing the process of solving the linear equation.