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Chapter 12: Problem 41

Solve each equation for \(y\). $$ x=-2 y-7 $$

Short Answer

Expert verified
The solution is \( y = -\frac{x}{2} - \frac{7}{2} \).

Step by step solution

01

Isolate the Term with y

Start with the equation given: \[ x = -2y - 7 \]. To isolate the term containing \( y \), we need to get rid of the constant on the right side. Add 7 to both sides of the equation:\[ x + 7 = -2y \].
02

Solve for y

Now that the term \( -2y \) is isolated, divide both sides of the equation by -2 to solve for \( y \):\[ y = \frac{x + 7}{-2} \]. This simplifies to:\[ y = -\frac{x + 7}{2} \] or \[ y = -\frac{x}{2} - \frac{7}{2} \].

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Key Concepts

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

Isolate Variable
Isolating a variable in an equation is a crucial step in solving formulas and equations, particularly linear ones. This means getting the variable by itself on one side of the equation. In the example given, the task was to solve for \( y \) in the equation \( x = -2y - 7 \). To do this, we start by focusing on the term that contains \( y \).
  • First, identify the term with \( y \) and the constants or other terms separated by addition or subtraction.
  • In the equation \( x = -2y - 7 \), \(-7\) is a constant that we want to remove from the equation's side with \( y \).
  • By adding 7 to both sides, \( x + 7 = -2y \), the constant is effectively removed from the right side.
  • This leaves \(-2y\) isolated, which is a step closer to solving for \( y \).
By isolating the variable, we make further steps simpler, allowing clearer paths to finding the solution.
Linear Equations
Linear equations are equations that form straight lines when graphed. They take the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. Solving these types of equations often involves managing terms and variables with simplified operations that maintain the equality.
  • In our example \( x = -2y - 7 \), we can recognize this as a linear equation between \( x \) and \( y \).
  • Solving such an equation involves basic operations like addition, subtraction, multiplication, or division to find the isolated variable's value.
  • Linear equations usually have one clear solution where both sides of the equation balance.
Understanding linear equations helps solve a wide range of mathematical problems and allows you to apply similar methods to more complex expressions.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to work with. This involves using basic algebraic operations to rearrange terms. In solving for \( y \) in \( x = -2y - 7 \), several manipulations were necessary:
  • After isolating the \(-2y\) term by adding 7 to both sides, the equation became \( x + 7 = -2y \).
  • To solve for \( y \), divide each side of the equation by \(-2\), which balances the operations across the equation.
  • This step transforms the equation into \( y = \frac{x + 7}{-2} \).
  • Further simplification gives us \( y = -\frac{x}{2} - \frac{7}{2} \), which splits the negative coefficient individually to both parts of the numerator.
Using algebraic manipulation techniques, such as inverse operations and simplification, helps achieve the solution effectively and keeps the equation balanced throughout the process.

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