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Chapter 3: Problem 48

Solve each equation. \(y-8=-5-1\)

Short Answer

Expert verified
The solution is \( y = 2 \).

Step by step solution

01

Simplify the Right Side of the Equation

We start by simplifying the right side of the equation. Combine the numbers:\(-5 - 1 = -6\). So, the equation simplifies to:\(y - 8 = -6\).
02

Isolate the Variable

To isolate \(y\), add \(8\) to both sides of the equation. This is done to eliminate the \(-8\) from the left side:\((y - 8) + 8 = -6 + 8\).This simplifies to:\(y = 2\).

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Key Concepts

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

Solving Linear Equations
The process of solving linear equations is all about finding the value of the variable that makes the equation true. A linear equation is an algebraic expression where each term is either a constant or the product of a constant and a single variable. An equation such as \(y - 8 = -5 - 1\) is called a linear equation because it can be transformed into a straight line on a graph.
  • Identify the equation to be solved.
  • Simplify each side if necessary. Combine like terms to make things clearer.
  • Apply operations to both sides to keep the equation balanced and solve for the variable.
Being methodical and performing operations step by step makes the process more straightforward. Remember, whatever you do to one side of the equation, do the exact same to the other side. That's the golden rule!
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves turning a more complex expression into a simpler one, while keeping its value the same. The aim is to make solving the equation easier. In the equation \(y - 8 = -5 - 1\), simplification happens on the right side by combining constant numbers:
  • Combine like terms: both \(-5\) and \(-1\) are constants, so we add them up, resulting in \(-6\). This makes the equation \(y - 8 = -6\).
Simplifying can also involve:
  • Distributing numbers across parentheses.
  • Combining variables of the same type, which means adding or subtracting coefficients.
Always look for parts of the equation or expression that can be combined, so that each term is as simplified as possible. This makes further operations less complicated.
Isolating Variables
Isolating a variable is a key step in solving equations, allowing us to find the variable's value. The goal is to get the variable \(y\) alone on one side of the equation.
In \(y - 8 = -6\), you've simplified as much as possible already. Now, you need to get \(y\) by itself.
  • To isolate \(y\), add \(8\) to both sides of the equation: \((y - 8) + 8 = -6 + 8\). This process will cancel out the \(-8\) on the left side.
  • Simplify both sides. On the left, \(-8 + 8\) equals zero, isolating \(y\). On the right, \(-6 + 8\) simplifies to \(2\).
Thus, the solution is \(y = 2\).
  • Perform operations that maintain the balance of the equation.
  • Focus on undoing whatever operations are interfering with the variable being isolated.
Using these steps ensures you're applying logical operations to uncover the value of \(y\) accurately, and with practice, this process can be swiftly applied to more complex problems.

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Most popular questions from this chapter

Solve each equation. \(\frac{y}{-6}=6-(-1)\)

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