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Chapter 6: Problem 77

Solve the equation. \(9=-\frac{3}{2}(x-2)\)

Short Answer

Expert verified
The solution to the equation is \(x = -4\).

Step by step solution

01

Multiply the fraction by the term in brackets

The equation can be rewritten as \(9 = -\frac{3}{2}x +3\) by distributing the \(-\frac{3}{2}\) to both terms inside the brackets.
02

Isolate x term

Subtract 3 from both sides to get the equation \(-\frac{3}{2}x = 6\).
03

Solve for x

Divide both sides by -\frac{3}{2}. We can multiply the both sides by the reciprocal of -\frac{3}{2} which is -\frac{2}{3}. The solution for the equation would be \(x=-\frac{2}{3}*6=-4\).

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Key Concepts

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

Distributive Property
The distributive property allows us to simplify expressions by distributing a multiplier to each term inside the parentheses. In the equation \(9 = -\frac{3}{2}(x-2)\), the distributive property is applied to the expression inside the parentheses.

Here the multiplier is \(-\frac{3}{2}\), and it multiplies both terms inside the brackets: \(x\) and \(-2\). This is how it transforms:
  • Multiply \(-\frac{3}{2}\) by \(x\), resulting in \(-\frac{3}{2}x\).
  • Multiply \(-\frac{3}{2}\) by \(-2\), resulting in \(+3\).
So, the equation \(9 = -\frac{3}{2}(x-2)\) simplifies to \(9 = -\frac{3}{2}x + 3\).
Understanding the distributive property is crucial for simplifying equations and expressions, especially when dealing with terms wrapped in parentheses.
Isolate Variable
To solve an equation for a specific variable, it is often necessary to isolate that variable on one side of the equation. In the context of our equation, we need to isolate \(x\).

Starting from the simplified equation \(9 = -\frac{3}{2}x + 3\), we want to get \(x\) by itself on one side. The first step is to eliminate the constant term on the side with \(x\).
This is achieved by subtracting 3 from both sides of the equation to remove it:
  • Subtract 3 from both sides: \(9 - 3 = -\frac{3}{2}x + 3 - 3\).
  • This results in: \(-\frac{3}{2}x = 6\).
Isolating the variable makes equations simpler to solve, as it allows us to directly address the variable we are solving for.
Reciprocal Multiplication
When we have a coefficient attached to our variable and need to isolate the variable completely, reciprocal multiplication can be a handy tool.

After isolating \(x\) in \(-\frac{3}{2}x = 6\), the goal is to get \(x\) on its own. The coefficient \(-\frac{3}{2}\) can be eliminated from \(x\) by multiplying both sides of the equation by its reciprocal.
The reciprocal of a fraction is just that fraction flipped, so the reciprocal of \(-\frac{3}{2}\) is \(-\frac{2}{3}\). Here's how it works:
  • Multiply both sides by \(-\frac{2}{3}\): \(-\frac{2}{3} \cdot (-\frac{3}{2}x) = 6 \cdot (-\frac{2}{3})\).
  • The left side simplifies as the reciprocals cancel out, leaving \(x\).
  • The right side simplifies to \(-4\).
Therefore, \(x = -4\).
Reciprocal multiplication is an effective method to solve for a variable in cases with fractional coefficients.

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