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Chapter 1: Problem 59

Solve each equation. $$2 k+30=180$$

Short Answer

Expert verified
The solution is \(k = 75\).

Step by step solution

01

Isolate the Variable Term

To solve the equation \(2k + 30 = 180\), first move the constant term to the other side of the equation by subtracting 30 from both sides. This gives: \(2k = 180 - 30\).
02

Simplify the Equation

Simplify the right-hand side of the equation: \(2k = 150\). This gives us a simpler equation where the variable term is isolated.
03

Solve for the Variable

To solve for \(k\), divide both sides of the equation by 2: \(k = \frac{150}{2}\).
04

Simplify the Solution

Simplify the division: \(k = 75\). This is the value of \(k\) that solves the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
When solving an equation, the first step is often to isolate the variable. This means you want to get the variable鈥攍ike \(k\) here鈥攐n one side of the equation and everything else on the other side. To do this, you use inverse operations: the opposite of what's been done to the variable. For instance, in the exercise \(2k + 30 = 180\), addition is used with 30. To "cancel out" this 30 and move it to the other side, you perform the inverse operation, which is subtraction. By subtracting 30 from both sides of the equation, you effectively detach the 30 from the variable term, which is exactly what you want.
Simplifying Equations
After isolating the variable term, the next step is simplifying the equation. Simplification makes equations easier to handle and understand. In our example, after subtracting 30 from both sides, you get \(2k = 150\). This step involves ensuring all numerical calculations are done. Here, the subtraction gives us a clearer view: you see that the original equation has now become leaner. Simplifying lets you focus on finding out what the variable, \(k\), truly represents. Remember that a simpler equation means fewer steps left until you find the solution!
Mathematical Operations
Mathematical operations are the basic building blocks that help in solving equations. They include addition, subtraction, multiplication, and division. In the equation \(2k + 30 = 180\), after isolating the variable and simplifying, the objective is to solve for \(k\). At this point, the equation \(2k = 150\) requires us to use division. Since \(2k\) means \(2 \times k\), you undo this by dividing both sides by 2. The operation simplifies the equation further, leading to \(k = 75\).
  • **Addition and subtraction** move terms from one side of the equation to the other.
  • **Multiplication and division** scale the variable up or down to solve it.
Understanding these operations is crucial because they allow you to navigate equations with confidence and ease.

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