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Chapter 2: Problem 41

Solve using the multiplication principle. Don't forget to check! $$ 5 x=70 $$

Short Answer

Expert verified
x = 14

Step by step solution

01

Isolate the Variable

To solve for the variable, we need to isolate it on one side of the equation. In this case, we have the equation: \[ 5x = 70 \]To isolate \(x\), divide both sides of the equation by 5.
02

Divide Both Sides by 5

When we divide both sides of the equation by 5, we get: \[ x = \frac{70}{5} \]Now, perform the division.
03

Simplify the Equation

Simplify the division: \[ x = 14 \]So, the value of \( x \) is 14.
04

Check the Solution

To ensure the solution is correct, substitute \( x = 14 \) back into the original equation: \[ 5(14) = 70 \]Calculate the left side: \[ 70 = 70 \]Since both sides are equal, the solution is verified.

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Key Concepts

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

multiplication principle
When solving linear equations, the multiplication principle is a key concept that simplifies solving equations. It states that if you multiply (or divide) both sides of an equation by the same non-zero number, the two sides remain equal.
In our example, we had the equation: \[ 5x = 70 \]We needed to isolate the variable x, so we used division (the inverse of multiplication) to achieve that. By dividing both sides by 5, we applied the multiplication principle in reverse to maintain balance in the equation.
isolating variables
Isolating the variable means getting the unknown variable by itself on one side of the equation. In our exercise, the equation was: \[ 5x = 70 \]To isolate x, we divided both sides of the equation by 5. This step makes x the subject of the equation, making it easier to solve.
Here鈥檚 a simple way to look at it:
  • If you have 5 apples (5x) and they equal to 70, divide both sides by 5 to find out how many apples (x) are in each group.
checking solutions
Checking the solution is an important step to verify that your solution is correct. After finding the value of x (which is 14 in this case), we substitute it back into the original equation to ensure it satisfies the equation: \[ 5(14) = 70 \]
Calculate the left-hand side: \[ 70 = 70 \]Since both sides of the equation are equal, our solution x = 14 is verified. This confirms the solution is correct and there were no mistakes made during the steps.
simplifying equations
Simplifying equations involves performing operations that make the equation easier to solve. In our example, we simplified the equation by dividing both sides by 5. Simplification reduced the problem to a straightforward division: \[ x = \frac{70}{5} \] which further simplifies to x = 14. Simplification is about making the equation less complex while preserving the balance between both sides.
If you follow these steps carefully鈥攊solating variables, applying the multiplication principle, and checking solutions鈥攜ou'll solve linear equations more effectively.

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