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Chapter 2: Problem 53

Solve each equation, and check the solution. $$ \frac{x}{7}=-5 $$

Short Answer

Expert verified
The solution is \(x = -35\).

Step by step solution

01

Isolate the Variable

To solve for the variable \(x\), multiply both sides of the equation by 7 to cancel out the denominator: \[ \frac{x}{7} \times 7 = -5 \times 7 \]
02

Simplify the Equation

Simplify both sides of the equation: \[ x = -35 \]
03

Check the Solution

Substitute \(x = -35\) back into the original equation to verify: \[ \frac{-35}{7} = -5 \]. Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

Isolating Variables
Isolating the variable is an essential first step in solving linear equations. The goal is to get the variable (in this case, \(x\)) by itself on one side of the equation. This often involves reversing operations that are currently applied to the variable.
In our example, \( \frac{x}{7} = -5 \), \(x\) is divided by 7. To isolate \(x\), we need to cancel out the division by 7. We do this by multiplying both sides of the equation by 7.
Here's what it looks like:
\[ \frac{x}{7} \times 7 = -5 \times 7 \] Multiplying both sides by the same number (in this case, 7) ensures the equation remains balanced. After performing this multiplication, we simplify to obtain: \[ x = -35 \]
  • Remember: isolating the variable usually involves reversing arithmetic operations like addition, subtraction, multiplication, or division.
  • Maintain the balance: whatever you do to one side of the equation, you must do to the other!
Multiplication Property of Equality
The multiplication property of equality states that if you multiply both sides of an equation by the same non-zero number, the two sides remain equal. This property is fundamental in solving linear equations as it helps to get rid of fractions.
In our example, after isolating the variable, we used this property to simplify the equation:
\[ \frac{x}{7} \times 7 = -5 \times 7 \] Applying the multiplication property of equality here allowed us to eliminate the fraction and directly solve for \(x\).
\[ x = -35 \] This step is crucial in transforming a more complex equation into a simpler one to solve.
Checking Solutions
After finding a solution, it's essential to check if it actually satisfies the original equation. This step helps confirm the solution's correctness and ensures no mistakes were made during the process.
Here's how to check the solution:
Substitute \(x = -35\) back into the original equation: \[ \frac{-35}{7} = -5 \] Simplify the left side: \[ -5 = -5 \] The left side equals the right side, which means our solution \(x = -35\) is correct.
  • Always double-check your solution by substituting it back into the original equation.
  • If both sides of the equation match, your solution is verified!
  • Checking solutions helps catch any potential mistakes during solving.
Double-checking, yay!

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