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Chapter 6: Problem 2

Solve and check each equation. \(x-3=-17\)

Short Answer

Expert verified
The solution to the equation \(x-3 = -17\) is \(x = -14\). Checked, the solution is correct.

Step by step solution

01

Isolate the variable \(x\)

Start with the given equation \(x-3 = -17\). Add 3 to both sides to isolate \(x\). This results in \(x = -17 + 3\).
02

Simplify the right side

Simplify the right side of equation. \(x = -17 + 3\) simplifies to \(x = -14\). So, the solution to the equation is \(x = -14\).
03

Check the solution

Substitute \(x = -14\) into the original equation to check the solution. Substitute \(x\) into \(x - 3 = -17\) gives \(-14 - 3\), which equals -17. Thus, -17 = -17. Since both sides of the equation are equal, the solution \(x = -14\) is correct.

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Key Concepts

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

Solving Equations
Solving equations involves finding the value of a variable that makes an equation true. An equation is a mathematical statement showing that two expressions are equal. For example, in the equation \(x - 3 = -17\), we need to find the value of \(x\) that makes both sides equal. To solve it, we usually use operations such as addition, subtraction, multiplication, or division to remove any terms from one side. The goal is to isolate the variable on one side to find its value. When you isolate the variable, you effectively "solve" the equation.
Variables
Variables are symbols that stand in for unknown values in mathematical expressions or equations. They are typically represented by letters such as \(x\), \(y\), or \(z\). In our exercise, the variable is \(x\), and our task is to determine its value. Variables allow us to construct general solutions and equations that can be used for a wide variety of situations. Understanding the role of variables is crucial as they signify the unknown part of the equation that needs to be discovered.
Simplification
Simplification is the process of making equations easier to solve by reducing them to their simplest form. This often involves combining like terms or performing arithmetic to condense an equation. In our example, after isolating \(x\), we simplify the equation \(x = -17 + 3\). We add \(3\) to \(-17\) to get \(-14\). Simplifying steps are vital as they often directly lead to the solution of the equation. By simplifying, we ensure that the equation is straightforward and easier to handle.
Equation Checking
Equation checking is the final step to verify our solution to an equation is correct. After finding a value for the variable, we substitute it back into the original equation. If both sides of the equation equal each other with this substitution, the solution is correct. In the exercise, we substitute \(x = -14\) back into the original equation, \(x - 3 = -17\), resulting in \(-14 - 3 = -17\), verifying our solution is correct. Checking is important to ensure that no errors were made in previous steps and the solution actually satisfies the equation.

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Most popular questions from this chapter

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}+8 x+12=0\)

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