How To Solve 3x + 2 = 15
===========================================================
Introduction
Solving linear equations is a fundamental concept in mathematics that helps us find the value of a variable in an equation. In this article, we will focus on solving the equation 3x + 2 = 15, which is a simple linear equation. We will break down the solution into step-by-step instructions, making it easy to understand and follow.
What is a Linear Equation?
A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants. In the equation 3x + 2 = 15, a = 3, b = 2, and c = 15.
Step 1: Subtract 2 from Both Sides
The first step in solving the equation 3x + 2 = 15 is to isolate the term with the variable (3x). To do this, we need to subtract 2 from both sides of the equation. This will give us:
3x + 2 - 2 = 15 - 2
Using the distributive property, we can simplify the left-hand side of the equation:
3x = 13
Step 2: Divide Both Sides by 3
Now that we have isolated the term with the variable (3x), we need to find the value of x. To do this, we need to divide both sides of the equation by 3. This will give us:
(3x) / 3 = 13 / 3
Using the quotient rule, we can simplify the left-hand side of the equation:
x = 13/3
Step 3: Simplify the Fraction
The fraction 13/3 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 13 and 3 is 1, so the fraction cannot be simplified further.
Step 4: Check the Solution
To ensure that our solution is correct, we need to plug it back into the original equation and check if it satisfies the equation. Let's substitute x = 13/3 into the original equation:
3(13/3) + 2 = 15
Using the distributive property, we can simplify the left-hand side of the equation:
13 + 2 = 15
This is indeed true, so our solution is correct.
Conclusion
Solving the equation 3x + 2 = 15 is a simple process that involves isolating the term with the variable, dividing both sides by the coefficient of the variable, and simplifying the fraction. By following these steps, we can find the value of x and check that it satisfies the original equation. This is a fundamental concept in mathematics that helps us solve linear equations and is essential for more advanced mathematical concepts.
Frequently Asked Questions
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form ax + b = c, where a, b, and c are constants.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the term with the variable, divide both sides by the coefficient of the variable, and simplify the fraction.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.
Additional Resources
Final Thoughts
Solving linear equations is a fundamental concept in mathematics that helps us find the value of a variable in an equation. By following the steps outlined in this article, you can solve linear equations and apply this knowledge to more advanced mathematical concepts. Remember to always check your solution by plugging it back into the original equation to ensure that it satisfies the equation.
=====================================================
Introduction
Linear equations are a fundamental concept in mathematics that help us solve problems involving variables. In this article, we will answer some of the most frequently asked questions about linear equations, providing a comprehensive guide to understanding and solving these equations.
Q: What is a Linear Equation?
A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form ax + b = c, where a, b, and c are constants.
Example:
2x + 3 = 5
In this equation, a = 2, b = 3, and c = 5.
Q: How Do I Solve a Linear Equation?
To solve a linear equation, you need to isolate the term with the variable, divide both sides by the coefficient of the variable, and simplify the fraction.
Step-by-Step Guide:
- Isolate the term with the variable.
- Divide both sides by the coefficient of the variable.
- Simplify the fraction.
Example:
Solve the equation 2x + 3 = 5.
- Isolate the term with the variable: 2x = 5 - 3
- Divide both sides by the coefficient of the variable: x = (5 - 3) / 2
- Simplify the fraction: x = 2 / 2 = 1
Q: What is the Difference Between a Linear Equation and a Quadratic Equation?
A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.
Example:
Linear Equation: 2x + 3 = 5
Quadratic Equation: x^2 + 2x + 1 = 0
Q: How Do I Solve a Quadratic Equation?
To solve a quadratic equation, you need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
Example:
Solve the equation x^2 + 2x + 1 = 0.
- Identify the values of a, b, and c: a = 1, b = 2, c = 1
- Plug the values into the quadratic formula: x = (-2 ± √(2^2 - 4(1)(1))) / 2(1)
- Simplify the expression: x = (-2 ± √(4 - 4)) / 2
- Simplify the square root: x = (-2 ± √0) / 2
- Simplify the expression: x = (-2 ± 0) / 2
- Simplify the expression: x = -2 / 2 = -1
Q: What is the Importance of Linear Equations?
Linear equations are important in mathematics because they help us solve problems involving variables. They are used in a wide range of fields, including physics, engineering, economics, and computer science.
Example:
Linear equations are used to model the motion of objects in physics. For example, the equation v = u + at models the velocity of an object under constant acceleration.
Q: How Do I Use Linear Equations in Real-Life Scenarios?
Linear equations are used in a wide range of real-life scenarios, including:
- Modeling the motion of objects in physics
- Solving problems in engineering and economics
- Creating algorithms in computer science
- Analyzing data in statistics
Example:
A company wants to model the cost of producing x units of a product. The cost is given by the equation C = 2x + 3, where C is the cost and x is the number of units produced.
Q: What are Some Common Mistakes to Avoid When Solving Linear Equations?
Some common mistakes to avoid when solving linear equations include:
- Not isolating the term with the variable
- Not dividing both sides by the coefficient of the variable
- Not simplifying the fraction
- Not checking the solution
Example:
Solve the equation 2x + 3 = 5.
- Isolate the term with the variable: 2x = 5 - 3
- Divide both sides by the coefficient of the variable: x = (5 - 3) / 2
- Simplify the fraction: x = 2 / 2 = 1
However, the solution x = 1 is not correct because it does not satisfy the original equation. The correct solution is x = 1, but the equation 2x + 3 = 5 is not satisfied when x = 1.
Conclusion
Linear equations are a fundamental concept in mathematics that help us solve problems involving variables. By understanding and solving linear equations, we can apply this knowledge to a wide range of fields, including physics, engineering, economics, and computer science. Remember to avoid common mistakes when solving linear equations, and always check the solution to ensure that it satisfies the original equation.
Additional Resources
Final Thoughts
Solving linear equations is a fundamental concept in mathematics that helps us solve problems involving variables. By following the steps outlined in this article, you can solve linear equations and apply this knowledge to a wide range of fields. Remember to always check the solution to ensure that it satisfies the original equation.