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Chapter 2: Problem 46

Solve each equation. Check all solutions. SEE EXAMPLE 4. (OB]ECTIVE 2) $$4 x+3=5 x$$

Short Answer

Expert verified
The solution to the given equation is \(x = 3\).

Step by step solution

01

Isolate the variable

The first step involves isolating the variable. Here, the variable is x. To isolate x, subtract 4x from both sides of the equation: \(4x - 4x + 3 = 5x - 4x\). This simplifies to \(3 = x\).
02

Solve the equation

After isolating the variable, the equation becomes very simple: \(3 = x\). Here, the value of the variable x seems to be 3.
03

Check the solution

Substitute x = 3 back into the original equation to check if it's a valid solution: Substitute x in the original equation \(4 x+3=5 x\) to give \(4 * 3 + 3 = 5 * 3\). Simplifying both sides gives 15 = 15, which is true, thus, the solution \(x = 3\) is found to be correct.

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, variables, often represented by letters like \( x \) or \( y \), stand in for numbers. These variables allow us to create equations, which are mathematical statements that show the relationship between different quantities.
  • An equation can usually be recognized by the '=' sign.
  • In algebra, we solve equations to find the value of the unknown variable.
Algebraic equations can represent real-world problems, making algebra a practical tool for problem-solving. As you become familiar with algebraic rules and operations, you learn to solve more complex equations. This foundational skill is crucial not only in advanced mathematics but also in various scientific and engineering fields.
Isolating Variables
Isolating the variable is a key step in solving linear equations. The goal is to get the variable alone on one side of the equation. This helps you determine the exact value of the variable. Linear equations are called 'linear' because they represent a straight line when graphed.
  • To isolate a variable, you need to perform operations that "undo" what has been done to the variable.
  • For example, if a variable is being added to a number, you should subtract that number from both sides of the equation.
  • If the variable is multiplied by a number, you'd divide both sides of the equation by that number.
In our example: First, subtract \( 4x \) from both sides to help isolate the variable \( x \). This leaves \( 3 = x \), which means the equation is solved for \( x \). Remember, the operations you do to one side of the equation must be done to the other side as well to maintain balance.
Checking Solutions
Once you've found a potential solution to an equation, it's important to check that solution to make sure it's correct. Checking solutions ensures that the value you've found truly satisfies the original equation.
  • To check a solution, substitute the value back into the original equation.
  • Simplify both sides of the equation to see if they are equal.
In this exercise, after isolating \( x \) and solving \( 3 = x \), you substitute \( x = 3 \) back into the original equation \( 4x + 3 = 5x \). Both sides simplify to 15 (\( 4 \times 3 + 3 \) and \( 5 \times 3 \)), confirming that \( x = 3 \) is indeed the correct solution. Verifying your solution is a crucial step in problem-solving. It boosts your confidence in your answer and illuminates any errors you may have made during your calculations.

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