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Chapter 2: Problem 30

In Exercises 27-32, solve the equation. $$ 10+x=3 $$

Short Answer

Expert verified
The solution to the equation \(10+x=3\) is \(x=-7\).

Step by step solution

01

Isolation of the Variable

At present the equation is \(10 + x = 3\), to isolate \(x\), the first step is to move \(10\) from the left side to the right side of the equation. This is carried out by subtracting \(10\) from both sides of the equation as it is an inverse operation to addition.
02

Subtract and Simplify

Subtract \(10\) from both sides of the equation, this results in \(x = 3-10\). Now, simplify this equation.
03

Evaluate the Expression

Performing the subtraction, \(3 - 10\), gives \(x = -7\).

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Key Concepts

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

Isolation of Variables
Isolation of variables is a critical concept when solving equations. It involves reorganizing an equation to get the unknown variable by itself on one side. This makes it easier to find the value of the variable. In the equation \(10 + x = 3\), our goal is to isolate \(x\).
You can achieve isolation by changing the structure of the equation. To do that, we'll need to move all terms involving the variable to one side and numbers to the other.
This concept is important because it allows you to solve for the variable effectively.
Inverse Operations
Inverse operations are operations that reverse the effect of the original operation. In our exercise, we are dealing with addition in the equation \(10 + x = 3\).

To isolate \(x\), you use subtraction, which is the inverse of addition, to move \(10\) from the left side to the right side of the equation. The equation becomes \(x = 3 - 10\).
  • Addition and subtraction are inverse operations.
  • Multiplication and division are also inverse operations.
This approach allows for simplifying the equation by cancelling out terms and isolating the variable.
Equation Simplification
Equation simplification is about reducing an equation to its simplest form. After using inverse operations to isolate a variable, you often need to simplify terms.
For our equation \(x = 3 - 10\), simplify the right side for easier computation. Simplifying reduces errors and speeds up finding the solution.
  • Simplification often involves combining like terms.
  • It helps make complex equations manageable.
Simplification is especially useful when equations become more complicated, ensuring clarity and correctness.
Integer Operations
Integer operations involve performing arithmetic operations with whole numbers, including positive and negative numbers.
In the equation \(x = 3 - 10\), you perform a subtraction operation with integers. Here, you compute \(3 - 10\), which equals \(-7\).
Understanding integer operations is vital because it affects the accuracy of your solutions. Remember these tips:
  • Adding a negative is equivalent to subtraction.
  • Subtracting a negative is like adding a positive.
  • Mastery of integer operations aids in solving diverse mathematical problems.
Building a strong foundation in integer operations enables you to handle various arithmetic challenges.

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Most popular questions from this chapter

The dimensions of a rectangular lawn are 150 feet by 250 feet. The property owner buys a rectangular strip of land \(x\) feet wide along one 250 -foot side of the lawn. Draw diagrams representing the lawn before and after the purchase. Write an expression for the area of each.

In Exercises \(1-10\), determine whether each value of \(x\) is a solution of the equation. \(7 x+1=4(x-2)\) (a) \(x=1\) (b) \(x=12\)

In Exercises 45-50, an expression for the balance in an account is given. Use a guess, check, and revise strategy to determine the time (in years) necessary for the investment of \(\$ 1000\) to double. Interest rate: \(6 \%\) $$ 1000(1+0.06)^{t} $$

In Exercises 27-32, solve the equation. $$ 3 x=30 $$

In Exercises 79-84, evaluate the expression. $$ |4(-6)(5)| $$
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