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Chapter 5: Problem 35

Solve and check. $$z+9=2$$

Short Answer

Expert verified
Z equals -7.

Step by step solution

01

Equation Setup

The given equation to solve is \(z + 9 = 2\). The goal is to isolate 'z' on one side of the equation to find its value.
02

Isolate the Variable

To isolate 'z', subtract 9 from both sides of the equation. This gives the new equation \(z = 2 - 9\).
03

Solve for 'z'

Solve the equation \(z = 2 - 9\) to find that \(z = -7\).
04

Check the Solution

To check this solution, substitute -7 for 'z' in the initial equation, \(z + 9 = 2\). Simplifying gives the equation \(-7 + 9 = 2\), which simplifies to \(2 = 2\). This confirms that ihe solution is valid.

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Key Concepts

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

Isolating Variables
When working with linear equations, one fundamental goal is isolating the variable. This means manipulating the equation so that the variable we want to solve for stands alone on one side of the equality sign. In our exercise, the equation is set up as z + 9 = 2. To isolate the variable z, we perform inverse operations, which in this case involves subtracting 9 from both sides. Here's a straightforward breakdown:

  • Original equation: z + 9 = 2
  • Subtract 9 from both sides: z + 9 - 9 = 2 - 9
  • Simplified equation: z = -7
This process of subtraction is a clear example of maintaining the balance of an equation, a core principle in algebra. By performing the same operation on both sides, we ensure that the equality holds true.
Equation Checking
Once you believe you've solved the equation, it's important to ensure the solution is correct. This verification process is known as equation checking. Take your proposed solution and plug it back into the original equation, and then simplify. In our case, we propose that z = -7. Plugging it back in:

  • Substitute -7 for z: -7 + 9 = 2
  • Simplify: 2 = 2
If after simplification, the equation holds true (as it does here), your solution is correct. If not, a mistake was made in the initial solving process, and you should revisit the beginning steps.
Basic Algebra
The foundation of solving linear equations lies within basic algebra, which involves understanding variables, constants, and operations. In the context of our exercise:

  • A variable, like z, represents an unknown value we aim to find.
  • A constant is a known value, like the 9 or 2 in our equation, which doesn't change.
  • Operations include addition, subtraction, multiplication, and division, which we use to manipulate the equation.
In our example, we use subtraction to move the constant to the other side of the equation, thereby isolating the variable. Basic algebra is much like a balancing act where whatever we do to one side of the equation, we must do to the other to maintain equality.

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Most popular questions from this chapter

Complete each statement with the word even or odd. If \(m\) is an even integer and \(n\) is an odd integer then \(m-n\) is an______ integer.

Make up an equation of the form \(a x=b\) that has \(-2\) as a solution.

Two joggers start at the same time from opposite ends of an 8 -mile jogging trail and begin running toward each other. One jogger is running at a rate of \(5 \mathrm{mph}\), and the other jogger is running at a rate of 7 mph. How long, in minutes, after they start will the two joggers meet?

Complete each statement with the word even or odd. If \(m\) and \(n\) are odd integers then \(m n\) is an______ integer.

Solve and check. $$7=\frac{2 x}{5}+4$$
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