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Chapter 5: Problem 26

Solve the equations. $$ \frac{a}{-3}=-17 $$

Short Answer

Expert verified
Answer: The value of 'a' in the given equation is 51.

Step by step solution

01

Identify the equation given

The given equation is: $$\frac{a}{-3}=-17$$
02

Multiply both sides by the denominator

In order to isolate 'a' and remove the denominator, we need to multiply both sides of the equation by -3. This will cancel out the denominator on the left side and give us a new equation: $$(-3) \cdot \frac{a}{-3} = (-17) \cdot (-3)$$
03

Simplify and solve for 'a'

Simplifying the equation from step 2, we are left with: $$a = 51$$ Thus, the value of 'a' is 51.

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a technique used to rearrange and simplify algebraic expressions or equations. It is critical for solving equations efficiently. When manipulating equations, we perform the same operation on both sides. This keeps the equation balanced.
  • First, identify terms on either side of the equation.
  • Apply operations like addition, subtraction, multiplication, or division.
  • Simplify the resulting expressions to get closer to the solution.

In the equation \( \frac{a}{-3} = -17 \), correct algebraic manipulation involves clearing the fraction by multiplying by the denominator, as performed in the steps given. Remember, the goal of manipulation is to make the equation as simple as possible.
Isolation of Variable
Isolation of variable is a method used to solve for a single variable, allowing us to find its value. The primary aim is to have one side of the equation with the variable alone. To do this, we eliminate other components step by step.

In the provided equation \( \frac{a}{-3} = -17 \), the variable \( a \) is part of a fraction. To isolate \( a \), you must remove the division by \(-3\). This requires multiplying both sides by the denominator, as shown. The multiplication cancels out the denominator on the left side, leaving the variable freestanding.

After elimination of all other terms, the variable is "isolated," making it easier to compute its value directly.
Multiplication of Equations
Multiplying equations is a tool used in algebraic manipulation to eliminate fractions or distribute terms. This technique is vital when dealing with equations that contain fractions.

Consider \( \frac{a}{-3} = -17 \). By multiplying both sides by \(-3\), you effectively cancel out the denominator on the left side. This leaves you with the simplified equation \( a = 51 \). The key here is to apply the same multiplication to both sides. This keeps the equation balanced and ultimately helps isolate the variable.
  • Choose the common multiple (often a denominator) to eliminate fractions.
  • Multiply both sides evenly to maintain balance.
  • Simplify each side step by step to solve the equation.

This approach is very effective in solving linear equations and provides a pathway to achieve the solution efficiently.

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