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Chapter 2: Problem 52

Solve each equation, and check the solution. $$ \frac{x}{5}=15 $$

Short Answer

Expert verified
x = 75; Solution verified.

Step by step solution

01

Title - Isolate the variable

Multiply both sides of the equation by 5 to isolate the variable x.
02

Title - Simplify the equation

Simplifying by multiplying both sides, we get: \(\frac{x}{5} \times 5 = 15 \times 5 \)
03

Title - Calculate the result

Perform the multiplication on the right side: 15 times 5 equals 75. So, \(\text{x} = 75\)
04

Title - Check the solution

Substitute x = 75 back into the original equation to confirm: \(\frac{75}{5} = 15 \). Since both sides 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 the Variable
Isolating the variable is the first step when solving linear equations. This means you want to get the variable (in this case, x) by itself on one side of the equation.
To do this, you perform operations that will eliminate any numbers or coefficients attached to the variable.
In the example \(\frac{x}{5}=15\), we multiply both sides by 5 to isolate x. This gives us:
\[ x = 15 \times 5 \] By isolating the variable, we've made it simpler to solve the equation.
Simplifying Equations
Simplifying equations is about making the math as straightforward as possible. Once you've isolated the variable, you need to simplify the resulting equation.
In this example, multiplying both sides of \(\frac{x}{5} = 15\) by 5 simplifies it to: \[ \frac{x}{5} \times 5 = 15 \times 5 \]
Carrying out the multiplication, we get: \[ x = 75 \]
Simplifying can take many forms, such as combining like terms or reducing fractions. The goal is to make the equation easier to work with.
Checking Solutions
After you've found a solution, it's crucial to verify it. This ensures there were no mistakes in your calculations.
To check the solution, substitute the value back into the original equation and see if it makes a true statement.
For our example, substitute x = 75 back into \(\frac{x}{5} = 15\): \[ \frac{75}{5} = 15 \]
This simplifies to 15 = 15, which is true. This verifies that x = 75 is the correct solution.
Basic Algebra
Basic algebra involves understanding and manipulating mathematical symbols and equations to solve problems.
Key concepts include variables (symbols representing numbers), coefficients (numbers multiplying variables), and the various operations (addition, subtraction, multiplication, division).
In our exercise, understanding these basics allows us to manipulate \(\frac{x}{5} = 15\) step by step:
  • We identify the variable x.
  • Recognize that x is divided by 5.
  • Use multiplication to undo the division.
Mastering basic algebra makes solving equations more intuitive and manageable.

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Most popular questions from this chapter

As of \(2017,\) the combined population of China and India was estimated at 2.7 billion. If there were about 0.9 as many people living in India as China, what was the population of each country, to the nearest tenth of a billion? (Data from U.S. Census Bureau.)

The Toyota Camry and the Honda Civic were the top-selling passenger cars in the United States in \(2016 .\) Civic sales were 22 thousand less than Camry sales, and 756 thousand of the two cars were sold. How many of each car were sold? (Data from wwy, businessinsider,com

The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In triangle \(A B C,\) the measure of angle \(A\) is \(141^{\circ}\) greater than the measure of angle \(B\). The measure of angle \(B\) is the same as the measure of angle \(C\). Find the measure of each angle.

During the 115 th session, the total number of Democrats and Republicans in the U.S. House of Representatives was \(434 .\) There were 46 more Republicans than Democrats. How many members of each party were there? (Data from www.house.gov)

When \(\frac{2}{3}\) of a number is added to \(10,\) the result is 5 more than the number. Find the number.
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