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Chapter 2: Problem 20

Solve each equation. Check all solutions. SEE EXAMPLE 1. (OBJECTIVE 1) $$5 x+3=8$$

Short Answer

Expert verified
The solution to the equation \( 5x + 3 = 8 \) is \( x = 1 \).

Step by step solution

01

Isolate the term with \( x \)

First, the goal is to isolate the term with \( x \) on one side of the equation. That can be achieved by subtracting 3 from both sides of the equation. It will then look like this: \( 5x + 3 - 3 = 8 - 3 \), which gives the simplified form: \( 5x = 5 \)
02

Solve for \( x \)

Now that the equation is \( 5x = 5 \), solving for \( x \) can be done by dividing both sides by 5: \( \frac{5x}{5} = \frac{5}{5} \). This leaves us with the solution: \( x = 1 \)
03

Check the solution

Lastly, it's important to check the solution by substituting \( x = 1 \) back into the original equation and confirming that it holds true. Substituting \( x = 1 \) into the original equation \( 5x + 3 = 8 \) yields \( 5*1 + 3 \), which equals 8. The right-hand side was the same as the left-hand side, making the solution correct.

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Key Concepts

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

Solving Linear Equations
Finding a solution for a linear equation means that you're trying to determine the value of a variable that makes the equation true. For single variable linear equations like \(5x + 3 = 8\), you will follow a series of steps to simplify and solve the equation.
  • Begin by identifying the variable you need to solve for, often represented as \(x\).
  • Use operations like addition, subtraction, multiplication, or division to isolate \(x\) on one side of the equation.
  • Ensure your operations maintain balance, meaning that what you do to one side, you must do to the other.

You can make your calculations easier by remembering that solving equations is like a puzzle. Piece by piece, you remove unnecessary parts until \(x\) is alone. Remember, the goal is to unravel the equation step by step until the variable stands alone with a value.
Checking Solutions
Once you've found a potential solution, like \(x = 1\) in our original exercise, the next step is to ensure that it solves the equation correctly. This is where checking your solution becomes crucial.
Substitute the found value back into the original equation to verify your work. So, replace \(x\) with 1 in the equation \(5x + 3 = 8\).
  • Compute the left side: \(5\times1 + 3 = 8\).
  • If both sides of the equation are equal, then your solution is correct.
  • If they are not equal, reconsider your calculations for possible mistakes.

Checking solutions is important because it confirms that your steps were correct. If your solution doesn't work, something could have gone wrong during your calculations, such as an arithmetic error.
Equation Isolation
Breaking down an equation to get the variable by itself is often referred to as equation isolation. This process is the heart of solving linear equations.
  • Start by identifying the term involving the variable, like \(5x\).
  • Remove other terms on the same side (for example, constants such as 3) using inverse operations like subtraction.
  • In the example \(5x+3=8\), you subtract 3 from both sides to get \(5x=5\).
By focusing on isolating the variable, you strategically simplify the equation. Isolation involves balancing both sides until the unknown is clear and explicit. Use these logical steps to approach any linear equation methodically, making the complex seem straightforward. Once isolated, the variable's value can be solved easily.

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