Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$3 x-5=2 x+7$$

Step-by-Step Solution

Answer

The solution to the equation \(3x - 5 = 2x + 7\) is \(x = 12\).

Step 1: Rearrange the equation

Let's start by moving the \(x\) terms to one side and the numbers to the other side. We can subtract \(2x\) from both sides and add \(5\) to both sides to achieve this. \n The equation becomes: \(3x - 2x = 7 + 5\). Simplified, this results in: \(x = 12\).

Step 2: Check the solution

We need to make sure that \(x = 12\) is indeed the solution to the original equation \(3x - 5 = 2x + 7\). Plug \(x = 12\) into the original equation and check if both sides are equal: \(3*12 - 5 = 2*12 + 7\), simplifying we find that \(36 - 5 = 24 + 7\). Further simplification, we observe that \(31 = 31\). Hence, \(x = 12\) is indeed the solution to the equation.

Key Concepts

Algebraic Equations

An algebraic equation is a mathematical statement that asserts the equality of two expressions. It typically includes variables, coefficients, constants, and the operations that connect them, such as addition, subtraction, multiplication, and division. In algebra, one of the most fundamental skills is learning how to solve these equations for the variable, which represents an unknown value.

For instance, in the algebraic equation \(3x - 5 = 2x + 7\), \(x\) is the variable. The goal when solving such an equation is to isolate the variable on one side to find its value. This process involves applying the same operation to both sides of the equation to maintain the equality, which is known as the balance method. The balance method is a visual way of understanding that what you do to one side of the equation, you must also do to the other side to keep it balanced.

Equation Solving Steps

The process for equation solving involves a series of logical and analytical steps, guiding you from the original problem to its solution. The steps are usually sequential, and following them is essential to solve the equation correctly and efficiently.

Identifying Like Terms

Initially, you look for like terms on both sides of the equation that can be simplified.

Moving Variables

Then, you move variables to one side of the equation and constants to the other to isolate the variable.

Simplification

The next step is to simplify both sides of the equation, if necessary. This might involve combining like terms or simplifying complex expressions.

Finding the Solution

Finally, you solve for the variable, finding its value. In our example \(3x - 2x = 7 + 5\), after simplification, you find \(x = 12\).

Solution Verification

Once you have arrived at a potential solution, verification is a critical next step. Verification ensures that the value you have determined for the variable actually satisfies the original equation. To verify the solution, you substitute the variable in the equation with the value found and simplify both sides to see if they are equal.

This step acts as a check and balance for your work, catching any possible errors in arithmetic or logic. In the given equation, when we substitute \(x = 12\) into \(3x - 5\) and \(2x + 7\), and find both sides equal \(31\), we affirm that \(x = 12\) is the correct solution to \(3x - 5 = 2x + 7\).

Ignoring this step can lead to false confidence in an incorrect solution, so always make sure to verify your results for accuracy. Finding that the equation balances with the solution \(x = 12\) gives us that assurance that the equation has been solved correctly.