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Chapter 1: Problem 45

Solve the equation. $$ -6 x-13=-x-23 $$

Short Answer

Expert verified
So, the solution is x = 2.

Step by step solution

01

Remove '-x' term from right hand side

We add 'x' on both sides of the equation to preserve the equality.\n So, -6x + x -13 = -23. Simplify, that gives us -5x - 13 = -23.
02

Remove constant from left hand side

Next, we add 13 to both sides of the equation to eliminate -13 from left side. Thus, -5x = -23 + 13. Simplify, it becomes -5x = -10.
03

Solve for x

Lastly, we divide both sides by -5 to find the value of x, x = -10/(-5) = 2.

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Key Concepts

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

Solving Equations
When we talk about solving equations, we mean finding the value of the variable that makes the equation true. In linear equations, we often have terms involving the variable and constant numbers.
To solve, we need to work methodically to simplify and rearrange the equation.
In this exercise, our goal is to find the value of \( x \) that satisfies the equation \(-6x - 13 = -x - 23\). We should change the structure to make the variable stand alone on one side.
  • Start by manipulating the terms to maintain a balanced equation.
  • This means doing the same mathematical operation on both sides.
  • The ultimate objective is to figure out what \( x \) equals, so it fulfills the given equation.
Equations can vary greatly in complexity, but by using logical and sequential steps, we can solve many types efficiently.
Equation Simplification
Equation simplification involves making an equation easier to work with by combining like terms and removing unnecessary elements. Simplification is a foundational skill in algebra that helps make more complex problems manageable.
In our example, we begin with the equation \(-6x - 13 = -x - 23\). The first step is to simplify the equation by combining like terms.
  • Move terms with \( x \) to one side by adding \( x \) to both sides, which gives us \(-5x - 13 = -23\).
  • Next, eliminate the constant on one side by adding or subtracting its opposite, aligning numbers only on one side (\(-5x = -10\)).
This strategic restructuring through simplification is a critical part of the problem-solving process.
Isolating Variables
Isolating variables means rearranging an equation to get the unknown variable alone on one side.
This technique is essential in solving equations because it directly gives us the variable's value.
Once we have simplified to \(-5x = -10\), we isolate \( x \) by dividing both sides by \(-5\).
  • This operation gives \( x = 2 \), which is the solution to the equation.
  • Division is chosen here because \( x \) is multiplied by \(-5\) and division is the inverse operation that eliminates the multiplier.
By isolating the variable, we identify the specific solution answering the original equation. This skill helps us solve not just algebraic problems but many other mathematical challenges as well.

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