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Chapter 4: Problem 77

solve the equation. (Lessons 3.1, 3.2, and 3.3) $$3(x-1)=-18$$

Short Answer

Expert verified
The solution of the equation is \(x = -5\)

Step by step solution

01

Expanding the Parentheses

First, apply the distributive property to expand the parentheses. This involves multiplying 3 by each term inside the parentheses: \(3 * x - 3 * 1 = -18\) After simplifying, the equation becomes: \(3x - 3 = -18\)
02

Isolating the x term

The next step is to isolate the x term, which means getting it alone on one side of the equation. This can be done by adding 3 to both sides of the equation: \(3x - 3 + 3 = -18 + 3\) This simplifies to: \(3x = -15\)
03

Solving for x

Finally, to solve for x, we divide both sides of the equation by 3: \(3x / 3 = -15 / 3\) which simplifies to: \(x = -5\)

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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 is used to simplify equations and expressions by distributing multiplication over addition or subtraction within parentheses. In this case, to solve the equation \(3(x-1)=-18\), we apply the distributive property. This means we multiply 3 by each term inside the parentheses. We perform:
  • Multiply 3 by \(x\), giving \(3x\).
  • Multiply 3 by -1, resulting in -3.
Now, replace \(3(x-1)\) with \(3x - 3\).
This step transforms our equation into \(3x - 3 = -18\).
Using the distributive property helps in eliminating parentheses, which simplifies the expression so we can focus on manipulating other parts of the equation.
Isolating Variables
After using the distributive property, our goal is to isolate the variable, which in this context is \(x\). Isolating the variable means getting \(x\) by itself on one side of the equation. We want the equation to have the form \(x = ...\) by the end.To achieve this, first eliminate any constant terms on the same side as the \(x\)-term. For the equation \(3x - 3 = -18\), we need to remove the \(-3\). We do this by adding 3 to both sides:
  • Add 3 to both sides: \(3x - 3 + 3 = -18 + 3\).
  • This simplifies to \(3x = -15\).
By isolating the variable \(x\), we make the equation simpler to solve, setting us up for the final step of solving for \(x\).
This process of managing terms and constants allows for a clearer pathway to the solution.
Solving for x
The last step in solving a linear equation is solving for \(x\), which means finding the value of \(x\) that makes the equation true. Now that we have isolated \(x\) in the equation \(3x = -15\), solving for \(x\) becomes straightforward.The equation involves a multiplication of \(x\) by 3, so we need to perform the inverse operation, which is division. Divide both sides of the equation by 3:
  • \(3x / 3 = -15 / 3\).
  • This simplifies to \(x = -5\).
By dividing both sides by 3, we cancel out the 3 next to \(x\), effectively solving the equation and finding \(x = -5\).
Solving for \(x\) at the final step provides the solution to the equation, giving us the value that satisfies the initial problem.

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