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Chapter 2: Problem 35

Solve each equation. $$ \frac{a}{-2}=1 $$

Short Answer

Expert verified
The solution is \( a = -2 \).

Step by step solution

01

Understand the Equation

The equation given is \( \frac{a}{-2} = 1 \). It indicates that when \( a \) is divided by \(-2\), the result is \(1\).
02

Isolate the Variable 'a'

To solve for \(a\), we need to eliminate the division by \(-2\). We can do this by multiplying both sides of the equation by \(-2\), the denominator of the fraction.
03

Multiply Both Sides by -2

Multiply both sides of the equation by \(-2\), which gives us: \( (-2) \times \frac{a}{-2} = 1 \times (-2) \). On the left side, the \(-2\) and \(-2\) cancel each other out, leaving us with \(a\).
04

Simplify the Equation

After multiplying, the equation simplifies to \(a = -2\). This means \(a\) is equal to \(-2\).

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Key Concepts

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

Division of Algebraic Expressions
In algebra, division is a fundamental operation used to simplify expressions and solve equations. When you have an algebraic expression like \( \frac{a}{-2} = 1 \), it means that the variable \( a \) is being divided by \(-2\). To solve such equations effectively, it's essential to understand the role of division:
  • Division helps reverse multiplication. This is crucial when isolating variables.
  • Think of division as spreading a value across a number of equal parts. In our case, \( a \) is spread over \(-2\) parts, resulting in \(1\).
  • Recognize that dividing by a negative number, like \(-2\), will influence the sign of the result.
To undo division in equations, you'll often use multiplication, which is the inverse operation. We'll explore this further as we isolate the variable and solve the equation.
Variable Isolation
Variable isolation is a key technique used to find the value of an unknown variable in an equation. In our context, isolating the variable \( a \) involves getting \( a \) by itself on one side of the equation:
  • The goal is to have the variable alone on one side, with everything else on the other side.
  • In \( \frac{a}{-2} = 1 \), \( a \) is tangled with \(-2\) through division. To "free" it, we need to get rid of the division by \(-2\).
  • This requires using operations inversely, like using multiplication when dealing with division.
Successfully isolating \( a \) helps clearly determine its value, simplifying the problem from a complex operation to just the variable and its value.
Multiplying Both Sides of an Equation
To solve equations like \( \frac{a}{-2} = 1 \), multiplying both sides by the same number is a powerful strategy. This technique ensures that the equation's balance is maintained:
  • Multiplication can reverse division. To remove \(-2\) from \( \frac{a}{-2} \), multiply both sides by \(-2\).
  • The negative numbers cancel out on the left side: \( (-2) \times \frac{a}{-2} \ = a \).
  • Always perform the same operation on both sides to maintain balance. If one side is multiplied, the other must be equally multiplied. Here, both sides are multiplied by \(-2\).
After multiplication, the equation simplifies to \( a = -2 \), showing how effective this method is in solving for the unknown variable.

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Most popular questions from this chapter

See Section 2.1 A plot of land is in the shape of a triangle. If one side is \(x\) meters, a second side is \((2 x-3)\) meters and a third side is \((3 x-5)\) meters, express the perimeter of the lot as a simplified expression in \(x .\) (GRAPH NOT COPY)

Solve. Five times a number, increased by one, is less than or equal to ten. Find all such numbers.

A recent survey showed that \(42 \%\) of recent college graduates named flexible hours as their most desired employment benefit. In a graduating class of 860 college students, how many would you expect to rank flexible hours as their top priority in job benefits? (Round to the nearest whole.) (Source: JobTrak.com)

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)

Explain how solving a linear inequality is similar to solving a compound inequality.
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