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Chapter 2: Problem 14

Solve each equation. \(-8 y-2=18\)

Short Answer

Expert verified
The solution is \(y = -\frac{5}{2}\).

Step by step solution

01

Add to Both Sides

To isolate the variable, start by eliminating the constant term on the left side of the equation. Add 2 to both sides of the equation. This will give:\[-8y - 2 + 2 = 18 + 2\]which simplifies to:\[-8y = 20\]
02

Divide Both Sides by Coefficient

Now, solve for \(y\) by dividing both sides of the equation by \(-8\), which is the coefficient of \(y\):\[\frac{-8y}{-8} = \frac{20}{-8}\]This simplifies to:\[y = -\frac{20}{8} = -\frac{5}{2}\]

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Key Concepts

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

Isolating Variables
Isolating variables is a fundamental skill when solving linear equations. It involves rearranging an equation so that the variable in question stands alone on one side of the equation, typically the left.
To do this, we often need to perform the same mathematical operation on both sides of the equation. In our given exercise \(-8y - 2 = 18\), we start by isolating the variable, here 'y', by first removing the constant term \(-2\), which shares the side with \(-8y\). This requires adding \(2\) to both sides of the equation, leaving us with a simpler equation \(-8y = 20\).
When isolating variables, remember:
  • Perform the same operation on both sides to maintain equality.
  • Always move constants and coefficients step-by-step, focusing on one action at a time.
Algebraic Manipulation
Algebraic manipulation allows us to adjust an equation to make it easier to solve. It involves using operations like addition, subtraction, multiplication, or division in strategic ways. These operations help in isolating variables or simplifying expressions. In the original problem, after isolating \(-8y\), we manipulated the equation by dividing both sides by \(-8\) because this is the coefficient of 'y'.
This manipulation transforms \(-8y = 20\) into \(y = -\frac{20}{8}\). Simplifying that fraction gives \(y = -\frac{5}{2}\).
Key tips for successful algebraic manipulation include:
  • Always execute the same operation on both sides of the equation to preserve balance.
  • Use inverse operations to cancel out terms quickly and efficiently.
Equation Simplification
Simplifying equations makes them easier to work with and understand. It's often one of the final steps in solving a linear equation. Simplification involves combining like terms and converting complex expressions into their simplest form.
In our equation, \(y = -\frac{20}{8}\), simplification means reducing the fraction to its simplest terms, \(-\frac{5}{2}\). This step is crucial as it provides a clear and final solution that is easy to interpret.
Here are some tips for simplifying equations:
  • Combine like terms whenever possible to reduce the number of components in an equation.
  • Remember to reduce fractions to their simplest form to make the solution clearer.
Equation simplification can appear as a small step, but it's essential for ensuring that solutions are accurate and easily interpreted.

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Most popular questions from this chapter

Solve each problem by setting up and solving an appropriate inequality. Find the solution set for each of the following compound statements, and in each case explain your reasoning. (a) \(x<3\) and \(5>2\) (b) \(x<3\) or \(5>2\) (c) \(x<3\) and \(6<4\) (d) \(x<3\) or \(6<4\)

Solve each equation and inequality. \(|3 x-1|-1=9\)

Solve each equation and inequality. \(\left|\frac{3 x-1}{4}\right|>3\)

Solve each compound inequality using the compact form. Express the solution sets in interval notation. \(-3<2-x<3\)

Solve each problem by setting up and solving an appropriate inequality. A person's intelligence quotient \((I)\) is found by dividing mental age \((M)\), as indicated by standard tests, by chronological age \((C)\) and then multiplying this ratio by 100 . The formula \(I=\frac{100 M}{C}\) can be used. If the \(I\) range of a group of 11-year-olds is given by \(80 \leq I \leq 140\), find the range of the mental age of this group.
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