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Chapter 1: Problem 16

Solve each equation and check your answer. $$5 x-3=-13$$

Short Answer

Expert verified
x = -2

Step by step solution

01

- Isolate the variable term

To isolate the term with the variable, add 3 to both sides of the equation: \[ 5x - 3 + 3 = -13 + 3 \] This simplifies to: \[ 5x = -10 \]
02

- Solve for the variable

Next, divide both sides by 5 to solve for \(x\): \[ \frac{5x}{5} = \frac{-10}{5} \] This simplifies to: \[ x = -2 \]
03

- Check the solution

Substitute \(x = -2\) back into the original equation to verify: \[ 5(-2) - 3 = -13 \] Calculate the left side: \[ -10 - 3 = -13 \] Since both sides of the equation are equal, the solution \(x = -2\) is correct.

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Key Concepts

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

Isolating the variable
Isolating the variable is the primary step in solving linear equations. The goal is to get the variable, usually expressed as 'x', by itself on one side of the equation. This is done by performing inverse operations on both sides of the equation. For the given exercise, we start with the equation \[5x - 3 = -13.\]\ The first step is to eliminate the constant term (-3) on the left side by adding 3 to both sides: \[5x - 3 + 3 = -13 + 3.\] This simplifies to \[5x = -10.\] The variable
Checking solutions
After finding a potential solution for the variable, it is crucial to check if it satisfies the original equation. This helps ensure our solution is accurate. For instance, once we find that \[x = -2\] from the prior steps, we substitute \[x = -2\] back into the original equation \[5x - 3 = -13. \]\ Here is the substitution: \[5(-2) - 3 = -13. \]\ Simplify the left-hand side: \[ -10 - 3 = -13.\] Since both sides of the equation are equal, the solution \[x = -2\] is confirmed to be correct. Always follow this verification step to be confident in your solutions.
Basic algebra steps
Understanding and applying basic algebra steps is essential for solving linear equations. Here are the fundamental steps condensed for clarity:
  • **Identify and simplify** - Look at the equation and simplify each side as much as possible.
  • **Isolate the variable** - Use addition, subtraction, multiplication, or division to get the variable term alone on one side of the equation. This often involves moving constants to the opposite side. In our example, this was adding 3 to both sides.
  • **Solve for the variable** - Once the variable is isolated, solve it by performing the necessary arithmetic operation on both sides. Here, we divided both sides by 5 to find \[x = -2.\]
  • **Check the solution** - Always plug the found variable back into the original equation to ensure both sides are equal. This confirms the solution's accuracy.
Mastering these steps helps to efficiently and accurately solve linear equations.

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Most popular questions from this chapter

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