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Chapter 3: Problem 47

Solve each equation. $$-2 y-10=5 y+18$$

Short Answer

Expert verified
The solution is \(y = -4\).

Step by step solution

01

Move All Terms to One Side

Start by moving all terms containing the variable \(y\) to one side of the equation. Subtract \(5y\) from both sides to isolate terms involving \(y\) on the left:\[-2y - 5y - 10 = 5y - 5y + 18\]This simplifies to:\[-7y - 10 = 18\]
02

Isolate the Term with the Variable

Next, eliminate the constant term from the left side by adding \(10\) to both sides:\[-7y - 10 + 10 = 18 + 10\]This gives:\[-7y = 28\]
03

Solve for the Variable

Finally, solve for \(y\) by dividing both sides by \(-7\):\[y = \frac{28}{-7}\]Calculate the division:\[y = -4\]

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Key Concepts

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

Prealgebra
Prealgebra serves as a foundation for understanding basic algebraic concepts and prepares you for more complex topics. It involves arithmetic operations and fundamental principles of algebra. When solving equations in prealgebra, some basic strategies are employed.
  • Using the same operation on both sides of the equation to maintain equality.
  • Combining like terms to simplify expressions.
  • Understanding negative and positive numbers.
In the original exercise, the goal is to solve for the variable. This involves ensuring the operations are applied correctly to both sides of the equation. Understanding the properties of equality is crucial in prealgebra, as what you do to one side of the equation, you must do to the other.
Linear Equations
Linear equations are mathematical statements that denote a straight line when graphed. Each linear equation represents the relationship between two variables without involving exponents or complicated functions. The equation from the exercise, \[-2 y - 10 = 5 y + 18\], is an example of a linear equation as it depicts a simple two-variable relationship without any non-linear components.
  • A linear equation has the general form: \(ax + b = cx + d\).
  • Solution strategies often include simplifying both sides to isolate variables.
  • Each operation must maintain the equation's balance.
It's important because solutions to linear equations can be represented visually on graphs as points where lines intersect, enhancing comprehension of both algebraic and geometric concepts.
Variable Isolation
Variable isolation is a pivotal technique in solving equations, especially linear equations. It means rearranging the equation so that the variable of interest stands alone on one side of the equation. Here鈥檚 how this is achieved:
  • Get all terms containing the variable on one side.
  • Simplify the equation to ensure the variable is by itself.
  • Use inverse operations to solve for the variable.
In our exercise, we start by simplifying the terms involving the variable \(y\) to one side, resulting in: \[-7y - 10 = 18\]. Then, we remove constant terms from the variable's side through addition or subtraction, proceeding to solve for \(y\) as demonstrated, ending with \(y = -4\). This process is essential in problem-solving as it allows you to isolate and find the exact value of the unknown variable.

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