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Chapter 1: Problem 22

Solve each equation, if possible. $$ 5 y+6=-18-y $$

Short Answer

Expert verified
y = -4

Step by step solution

01

- Combine like terms

First, add and subtract the terms involving y on both sides of the equation to move all the y-terms to one side. Starting equation: \[ 5y + 6 = -18 - y \] Add y to both sides: \[ 5y + y + 6 = -18 - y + y \] This simplifies to: \[ 6y + 6 = -18 \]
02

- Isolate the variable term

Next, move the constant term to the other side by subtracting 6 from both sides. Starting from: \[ 6y + 6 = -18 \] Subtract 6 from both sides: \[ 6y + 6 - 6 = -18 - 6 \] This simplifies to: \[ 6y = -24 \]
03

- Solve for y

Finally, solve for y by dividing both sides by 6. Starting from: \[ 6y = -24 \] Divide both sides by 6: \[ y = \frac{-24}{6} \] This simplifies to: \[ y = -4 \]

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Key Concepts

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

Combining Like Terms
When solving linear equations, combining like terms is one of the first and most important steps. Like terms are terms in an equation that have the same variable raised to the same power. In the context of the given equation, \[ 5y + 6 = -18 - y \], the terms involving 'y' are \[ 5y \] and \[ -y \]. Combining like terms makes the equation easier to solve because it reduces the number of terms in the equation.
To combine like terms, we added 'y' to both sides of the equation. This grouped all the \[ y \] terms on one side: \[ 5y + y + 6 = -18 - y + y \].This simplified to: \[ 6y + 6 = -18 \]. Combining like terms helps in reducing and simplifying the equation, making subsequent steps more straightforward.
Isolating Variable
After combining like terms, the next crucial step is isolating the variable. Isolating the variable means getting the variable 'y' alone on one side of the equation.
Starting with the simplified equation: \[ 6y + 6 = -18 \] , we move the constant term to the other side by performing the opposite mathematical operation. Here, we subtract 6 from both sides to isolate the \[ 6y \] term: \[ 6y + 6 - 6 = -18 - 6 \] . This simplifies to: \[ 6y = -24 \].
Isolating the variable is essential as it leaves us with a much simpler equation that can be solved in the next step. This process often involves addition, subtraction, multiplication, or division.
Basic Algebra
Understanding basic algebra is fundamental when solving linear equations. Basic algebra involves manipulating variables and constants to find the values of unknowns. Working through the equation \[ 5y + 6 = -18 - y \] , we used basic algebraic principles like combining like terms and isolating the variable to simplify the equation. Once the variable was isolated: \[ 6y = -24 \], basic algebra led us to solve for \[ y \] .
To solve \[ 6y = -24 \], we divided both sides by 6, a fundamental algebraic operation: \[ y = \frac{-24}{6} \], resulting in: \[ y = -4 \].
Mastering these basic algebraic operations is crucial as they form the building blocks for more complex mathematical concepts and are used frequently in various real-world applications.

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Most popular questions from this chapter

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