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Chapter 3: Problem 8

Solve each equation. $$-4 z-6=2(z+3)$$

Short Answer

Expert verified
So, the solution to the equation is \(z = -2\).

Step by step solution

01

Distribute Right-Hand Side

We start by distributing \(2(z+3)\) on the right-hand side. This yields \(2z + 6\). So, the equation becomes \(-4z - 6 = 2z + 6\).
02

Combine Like Terms

Let's gather z terms on one side of the equation and constant terms on the other. We subtract \(2z\) from both sides: \(-4z - 2z = 6 + 6\). The result of that operation shows \(-6z = 12\).
03

Solve for the Variable z

The final step is to isolate the variable z, by dividing the whole equation by \(-6\): \(-6z / -6 = 12 / -6\). Which finally gives: \(z = -2\).

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Key Concepts

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

Equation Solving
Equation solving can feel like a puzzle that involves rearranging pieces until the picture comes together. In mathematics, equations are like statements of equality and solving them means finding the value of the variable that makes the statement true.

Consider the equation \[-4z - 6 = 2(z + 3)\]. Here, the goal is to determine the value of \(z\) that satisfies this equality.
Remember, whatever changes you make in an equation, they should balance out on both sides. This is key to maintaining the equality and eventually "solving" the puzzle.
Distributing Terms
Distributing terms is one of the first steps in solving equations, especially when you encounter parentheses.
In the given equation, \(2(z + 3)\), we need to distribute the \(2\) across the terms inside the parentheses. This process involves multiplying the \(2\) by each term inside:
  • \(2 \times z = 2z\)
  • \(2 \times 3 = 6\)

After distribution, the equation transforms into:
\[-4z - 6 = 2z + 6\]
Distributing terms helps simplify the equation, making it easier to handle in later steps.
Combining Like Terms
Combining like terms is all about simplifying the equation by merging terms that have the same variable.
In our example, notice that \(-4z\) and \(2z\) are like terms because they both contain the variable \(z\).
Similarly, constants like \(-6\) and \(6\) should be combined on one side.
  • Move the \(2z\) to the left side by subtracting it from both sides: \(-4z - 2z\)
  • On the other side, add the constants: \(6 + 6\)

After combining like terms, we get: \(-6z = 12\). This simplifies the equation, bringing us closer to finding the value of the variable.
Solving for a Variable
The final step is solving for the variable, which means isolating it on one side of the equation.
We have \(-6z = 12\). To find \(z\), you want \(z\) by itself on the left side of the equation.
This is done by dividing both sides by \(-6\):
  • \(-6z / -6 = z\)
  • \(12 / -6 = -2\)

The division cancels out the \(-6\) on the left, leaving you with \(z = -2\).
The solution tells us that when \(z\) is \(-2\), the original equation holds true.

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Most popular questions from this chapter

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$A(-2, y+1) \text { and } B(6, y-1)$$

Express each inverse variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 2. (OBJECTIVE 2) \(J\) varies inversely with \(v .\) If \(J=90\) when \(v=5,\) find \(J\) when \(v=45\).

Give examples of an equation in one variable and an equation in two variables. How do their solutions differ?

Set up a variation equation and solve for the requested value. For a fixed area, the length of a rectangle is inversely proportional to its width. A rectangle has a width of 8 feet and a length of 10 feet. If the length is increased to 16 feet, find the width of the rectangle.

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(i\) varies inversely with \(d^{2}\). If \(i=8\) when \(d=3\), find \(i\) when \(d=6\).
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