Multiply or divide by a coefficient

Oct 13, 2025 by pinkandtrash.com 129 views

Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will focus on solving linear equations, specifically the equation 5a - 10 + a = 44. We will break down the solution step by step, using a combination of algebraic manipulations and logical reasoning.

Understanding the Equation

The given equation is 5a - 10 + a = 44. To solve for 'a', we need to isolate the variable 'a' on one side of the equation. The equation consists of two terms involving 'a', which are 5a and a. We also have a constant term, -10, and a constant value, 44.

Combining Like Terms

The first step in solving the equation is to combine like terms. In this case, we have two terms involving 'a', which are 5a and a. We can combine these terms by adding their coefficients. The coefficient of 5a is 5, and the coefficient of a is 1. Therefore, we can combine these terms as follows:

5a + a = 6a

Now, the equation becomes:

6a - 10 = 44

Adding 10 to Both Sides

The next step is to add 10 to both sides of the equation to eliminate the constant term. This will help us isolate the variable 'a' on one side of the equation. Adding 10 to both sides gives us:

6a - 10 + 10 = 44 + 10

Simplifying the equation, we get:

6a = 54

Dividing Both Sides by 6

Now, we need to divide both sides of the equation by 6 to solve for 'a'. This will give us the value of 'a' that satisfies the equation. Dividing both sides by 6 gives us:

6a / 6 = 54 / 6

Simplifying the equation, we get:

a = 9

Conclusion

In this article, we solved the linear equation 5a - 10 + a = 44 step by step. We combined like terms, added 10 to both sides, and finally divided both sides by 6 to solve for 'a'. The value of 'a' that satisfies the equation is 9. This solution demonstrates the importance of algebraic manipulations and logical reasoning in solving linear equations.

Tips and Tricks for Solving Linear Equations

Solving linear equations can be a challenging task, but with practice and patience, you can master the techniques. Here are some tips and tricks to help you solve linear equations:

Combine like terms: Combine terms involving the same variable to simplify the equation.
Add or subtract constants: Add or subtract constants to both sides of the equation to eliminate the constant term.
Multiply or divide by a coefficient: Multiply or divide both sides of the equation by a coefficient to isolate the variable.
Check your solution: Check your solution by plugging it back into the original equation to ensure that it satisfies the equation.

Real-World Applications of Linear Equations

Linear equations have numerous real-world applications in various fields, including:

Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and force.
Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and optimization problems.
Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Conclusion

In conclusion, solving linear equations is a fundamental skill in mathematics that has numerous real-world applications. By combining like terms, adding or subtracting constants, multiplying or dividing by a coefficient, and checking your solution, you can master the techniques of solving linear equations. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

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.

Q: How do I solve a linear equation?

A: To solve a linear equation, combine like terms, add or subtract constants, multiply or divide by a coefficient, and check your solution.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications in physics, engineering, economics, and computer science.

Q: How do I check my solution?

A: To check your solution, plug it back into the original equation to ensure that it satisfies the equation.

Introduction

Solving linear equations is a fundamental skill in mathematics that has numerous real-world applications. In our previous article, we provided a step-by-step guide to solving linear equations. In this article, we will answer some frequently asked questions about solving linear equations.

Q&A Guide

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, and x is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

Combine like terms: Combine terms involving the same variable to simplify the equation.
Add or subtract constants: Add or subtract constants to both sides of the equation to eliminate the constant term.
Multiply or divide by a coefficient: Multiply or divide both sides of the equation by a coefficient to isolate the variable.
Check your solution: Check your solution by plugging it back into the original equation to ensure that it satisfies the equation.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
Not checking your solution: Failing to check your solution can lead to incorrect answers.
Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.

Q: How do I check my solution?

A: To check your solution, plug it back into the original equation to ensure that it satisfies the equation. This can be done by substituting the solution into the equation and verifying that it is true.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications in:

Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and force.
Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and optimization problems.
Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Q: How do I graph a linear equation?

A: To graph a linear equation, follow these steps:

Find the x-intercept: Find the x-intercept by setting y = 0 and solving for x.
Find the y-intercept: Find the y-intercept by setting x = 0 and solving for y.
Plot the intercepts: Plot the intercepts on a coordinate plane.
Draw a line: Draw a line through the intercepts to represent the linear equation.

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.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, follow these steps:

Write the equations: Write the equations in the form ax + by = c.
Use substitution or elimination: Use substitution or elimination to solve the system of equations.
Check your solution: Check your solution by plugging it back into the original equations to ensure that it satisfies the equations.

Conclusion

Solving linear equations is a fundamental skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can master the techniques of solving linear equations and apply them to real-world problems.

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.

Q: How do I solve a linear equation?

A: To solve a linear equation, combine like terms, add or subtract constants, multiply or divide by a coefficient, and check your solution.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications in physics, engineering, economics, and computer science.

Q: How do I check my solution?

A: To check your solution, plug it back into the original equation to ensure that it satisfies the equation.

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.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, use substitution or elimination to solve the system of equations, and check your solution by plugging it back into the original equations to ensure that it satisfies the equations.