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Chapter 2: Problem 43

Solve the equations. More than one step is needed in each case.$$2 x-3=15$$

Short Answer

Expert verified
The solution is \(x = 9\).

Step by step solution

01

Isolate the Term with Variable

To solve the equation \(2x - 3 = 15\), the first step is to isolate the term that contains the variable. We begin by adding 3 to both sides of the equation: \[2x - 3 + 3 = 15 + 3\] Simplifying both sides gives us \(2x = 18\).
02

Solve for the Variable

Now that we have isolated the variable term, we need to solve for \(x\). We do this by dividing both sides by 2: \[\frac{2x}{2} = \frac{18}{2}\] Simplifying the division results in \(x = 9\).

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Key Concepts

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

Solving Equations
Solving equations is a fundamental aspect of algebra. This involves finding the value of the variable that makes the equation true. In the case of the equation \(2x - 3 = 15\), we aim to determine the value of \(x\) such that both sides of the equation equal the same number.

Completing this process often requires several steps, and understanding these steps can make solving equations easier. The idea is to perform operations on both sides of the equation to maintain balance, as if the equation is on a seesaw. If you add, subtract, multiply, or divide on one side, you must do the exact same thing on the other side.

Here are the basic steps for solving equations:
  • Identify the variable term and isolate it (we'll cover this in more detail later).
  • Switch operations鈥攊f a term is subtracted, add it to both sides, and if multiplied, divide both sides to solve for the variable.
  • Keep the equation balanced by performing operations equally on both sides.
Isolate the Variable
Isolating the variable is a key step in solving equations, especially multi-step ones. This involves getting the term with the variable alone on one side of the equation. By doing this, we simplify the problem, which makes it easier to solve for the variable.

In the example \(2x - 3 = 15\), the variable \(x\) is part of the term \(2x\). To isolate it:
  • First, we remove any constant terms from the side of the equation with the variable. In this case, by adding 3 to both sides, we eliminate \(-3\), simplifying the equation to \(2x = 18\).
  • This step effectively "clears" the equation, focusing solely on the variable and its coefficient.
Isolating variables helps establish the groundwork for the final steps of solving the equation, after which the specific value of the variable can be found.
Step-by-Step Solutions
Breaking down problems into step-by-step solutions is not only beneficial for understanding, but necessary for more complex equations. Each step is a logical progression that, when followed, leads to the correct answer.

In the given equation \(2x - 3 = 15\):
  • Step 1: We added 3 to both sides to eliminate \(-3\), bringing us to \(2x = 18\).
  • Step 2: We divided each side by 2 to solve for \(x\), resulting in \(x = 9\).
A step-by-step approach helps track each action taken to ensure the solution's accuracy. It also helps reinforce understanding of how variables behave under different operations. Following each step methodically reduces errors and builds a strong foundation for more advanced algebraic concepts later on.

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Most popular questions from this chapter

Apply the order of operations and answer the questions. The volume of a right rectangular pyramid with a height of 7.59 centimeters and a square base that is 2.43 centimeters on a side is given by \(\frac{2.43^{2} \cdot 7.59}{3} .\) Evaluate the expression and interpret the result. Round the volume to two decimal places.

Use the formula $$ V=\frac{1}{3} \pi\left(\frac{w}{2}\right)^{2} h $$ for the volume \(V\) of a right circular cone, where \(w\) represents the width of the base (this is the same as the diameter) and \(h\) represents the height. Round the results to two decimal places and include proper units. Salt that is used to treat icy roads and highways during the winter is stored in a cone-shaped building. Typically, these buildings are 65 feet wide and 35 feet tall. What is the volume of one of these salt storage buildings?

Solve the equations by using the multiplication property. $$\frac{y}{7}=-7$$

Apply the order of operations and answer the questions. The volume of a right rectangular pyramid with a height of 27.6 centimeters and a square base that is 22.1 centimeters on a side is given by \(\frac{22.1^{2} \cdot 27.6}{3} .\) Evaluate the expression and interpret the result. Round the volume to two decimal places.

Solve the equations by using the subtraction property. $$-9=x+3$$
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