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Chapter 1: Problem 7

Solve and check linear equation. \(7 x+4=x+16\)

Short Answer

Expert verified
The solution to the equation is \(x = 2\).

Step by step solution

01

Simplify both sides

First, simplify both sides of the equation by subtracting x from both sides. This will result in: \(7 x - x = 16 - 4\).
02

Simplify again

Now, simplify both sides of the equation once more: \(6 x = 12\).
03

Solve for x

Finally, solve for the variable x by dividing both sides of the equation by 6. This will result in: \(x = 2\).
04

Check the solution

Ensure the solution is correct by substituting \(x = 2\) back into the original equation: \(7*2+4=2+16\). When simplified: \(14 + 4 = 18\), so indeed \(18 = 18\). Thus, the solution is correct.

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Key Concepts

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

Algebraic Simplification
Algebraic simplification is the process of reducing an equation or expression to its simplest form. In solving linear equations, this often involves combining like terms or applying basic mathematical operations across terms on each side of the equation.

Consider the equation from our exercise:
  • Start with the equation: \(7x + 4 = x + 16\).
  • To simplify, perform operations to make both sides simpler. This often means eliminating terms that can be combined or moved.
  • In our case, subtract \(x\) from both sides to get \(7x - x + 4 = 16\), which simplifies to \(6x + 4 = 16\).
  • Next, subtract 4 from both sides to further simplify: \(6x = 12\).
These steps help streamline the equation, preparing it for solving, by reducing the number of terms and isolating the variable.
Checking Solutions
Once you have found a solution to an equation, it is crucial to check its correctness. This involves substituting the solution back into the original equation to verify that both sides are equal.

For the equation \(7x + 4 = x + 16\), let's check if \(x = 2\) is the correct solution:
  • Replace \(x\) with 2 in the original equation: \(7(2) + 4\) and \(2 + 16\).
  • Calculate each side separately: \(7 \times 2 + 4 = 14 + 4 = 18\) and \(2 + 16 = 18\).
  • Both sides equal 18, confirming the solution is correct.
Checking solutions ensures your algebraic manipulations were accurate and verifies the result matches the original conditions of the equation.
Variable Isolation
Variable isolation is a key step in solving algebraic equations, especially linear ones. The goal is to manipulate the equation such that the variable stands by itself on one side of the equation.

Here's how you might achieve this using our example:
  • Start with the simplified equation: \(6x = 12\).
  • To isolate \(x\), divide both sides by the coefficient of \(x\), which is 6.
  • Performing this division gives \(x = \frac{12}{6}\).
  • Simplify the fraction to solve for \(x\): \(x = 2\).
By using operations that keep the equation balanced, we successfully isolate \(x\).
This enables us to clearly see the value of the variable, leading to the solution of the original equation.

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