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Chapter 2: Problem 28

Solve each equation. $$ -y=8 $$

Short Answer

Expert verified
The solution is \( y = -8 \).

Step by step solution

01

Identify the Objective

The objective is to isolate the variable, in this case, \( y \), on one side of the equation to solve for \( y \).
02

Remove Negative Sign

To solve \( -y = 8 \), you need to remove the negative sign by multiplying both sides of the equation by \( -1 \).
03

Simplify Both Sides

After multiplying, the equation becomes \( y = -8 \). This isolates \( y \), giving us the value of the variable.

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Key Concepts

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

Isolation of Variables
The concept of isolating variables is straightforward once you understand its purpose. To solve for a variable means getting it alone on one side of the equation. In the equation \(-y = 8\), our objective is to solve for \(y\). This means we need \(y\) by itself on either the left or right side.
  • Look for the variable: It's important to identify which term you need to get by itself. In our case, it is \(-y\).
  • Perform operations to move other numbers or constants: You do this by using basic arithmetic operations, like addition, subtraction, multiplication, or division.
Understanding isolation of variables simplifies complex problems and lays the groundwork for tackling more challenging equations.
Negative Sign Multiplication
Multiplying to remove a negative sign is a crucial step in solving equations with negative coefficients. In the equation \(-y = 8\), the variable \(y\) is accompanied by a negative sign, which we can eliminate by multiplying the entire equation by \(-1\).
  • This basic operation flips the sign of each term in the equation, turning \(-y\) into \(y\).
  • Remember to do the same to both sides of the equation to maintain equality, hence \(8\) becomes \(-8\).
This technique ensures that our variable \(y\) is positively isolated, making it much easier to interpret the solution.
Simplifying Equations
Simplifying equations is all about making them as clean and simple as possible. After multiplying both sides of \(-y = 8\) by \(-1\), the equation simplifies to \(y = -8\). This accounts for all parts, ensuring the variable is isolated and the equation is as straightforward as possible.
  • Simplification helps in confirming that our steps are correct and that the equation balances out.
  • Every mathematical operation performed should lead toward an equation where the solution is crystal clear.
Simplifying makes complicated equations accessible and trains your problem-solving skills for more advanced math.

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Most popular questions from this chapter

At this writing, the women's world record for throwing a disc (like a heavy Frisbee) was set by Jennifer Griffin of the United States in \(2000 .\) Her throw was 138.56 meters, The men's world record was set by Christian Sandstrom of Sweden in 2002 . His throw was \(80.4 \%\) farther than Jennifer's. Find the distance of his throw. Round to the nearest meter. (Source: World Flying Disc Federation)

Solve each inequality. Graph the solution set and write it in interval notation. $$ -6<3(x-2) \leq 8 $$

Solve each inequality. Graph the solution set and write it in interval notation. $$ 0<4(x+5) \leq 8 $$

The total number of movie screens in the United States has been increasing in recent years. In 2005 , there were \(37,092\) in door movic screens. This is a \(4.3 \%\) increase from the number of indoor movie screens in \(2000 .\) How many movie screens were operating in \(2000 ?\) Round to the nearest whole. (Source: National Association of Theater Owners)

Explain how solving a linear inequality is different from solving a compound inequality.
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