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Chapter 2: Problem 27

Solve each equation, and check the solution. $$ -7 x=28 $$

Short Answer

Expert verified
The solution is \( x = -4 \).

Step by step solution

01

Isolate the Variable

To isolate the variable, divide both sides of the equation by -7. This will cancel out the -7 on the left side. \[ \frac{-7x}{-7} = \frac{28}{-7} \] Simplifying this, we get: \[ x = -4 \]
02

Check the Solution

Plug the solution back into the original equation to verify its correctness.Original equation: \[ -7x = 28 \] Substituting \( x = -4 \) into the equation: \[ -7(-4) = 28 \] This simplifies to \[ 28 = 28 \] which is true, so \( x = -4 \) is the correct solution.

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Key Concepts

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

Isolating the variable
When solving linear equations, the goal is to isolate the variable on one side of the equation. In our example, the equation is \$ -7x = 28 $$.
Here鈥檚 how you do it step-by-step:
  • Identify the term containing the variable. In this case, it's \$ -7x $$.
  • To isolate \$ x $$, we need to eliminate the coefficient \$ -7 $$ from the left side.
  • Divide both sides of the equation by \$ -7 $$: \$ \frac{-7x}{-7} = \frac{28}{-7} $$.
  • This simplifies to \$ x = -4 $$.
Isolating the variable is a critical step because it allows us to find the value of \$ x $$ that satisfies the equation.
By performing the same operation on both sides, we maintain the equality of the equation.
Checking solutions
After finding a solution, it's essential to verify its correctness. We do this by substituting the solution back into the original equation. For our example, we found \$ x = -4 $$.
Here鈥檚 how to check it:
  • Start with the original equation: \$ -7x = 28 $$.
  • Substitute \$ -4 $$ for \$ x $$: \$ -7(-4) = 28 $$.
  • This simplifies to \$ 28 = 28 $$, which is true.
Since both sides of the equation are equal, \$ x = -4 $$ is the correct solution. This step ensures that our solution is accurate and reliable.
Algebraic techniques
There are various algebraic techniques used to solve linear equations. Here, we focused on division.
Some key techniques include:
  • Distribution: Expanding expressions using the distributive property, e.g., \$ a(b + c) = ab + ac $$.
  • Combining like terms: Simplifying expressions by adding or subtracting like terms, e.g., \$ 2x + 3x = 5x $$.
  • Division and Multiplication: Isolating variables by using inverse operations, like in our example, dividing both sides by the variable's coefficient.
  • Substitution: Checking solutions by substituting variables back into the original equation.
Mastering these algebraic techniques allows for efficient and accurate solutions to linear equations.
Each technique is a tool that can be used depending on the structure of the equation.

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