Solve And Check The Linear Equation.${ 9y - 6 = 16y + 50 }$

Mar 9, 2025 by ADMIN 61 views

**Solving and Checking Linear Equations: A Step-by-Step Guide**

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving and checking linear equations, providing a step-by-step guide on how to approach these types of problems.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:

ax + b = c

where a, b, and c are constants, and x is the variable.

Example of a Linear Equation

Let's consider the following linear equation:

9y - 6 = 16y + 50

This equation is a linear equation because the highest power of the variable (y) is 1.

Solving Linear Equations

To solve a linear equation, we need to isolate the variable on one side of the equation. We can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Step 1: Simplify the Equation

The first step in solving a linear equation is to simplify the equation by combining like terms. In the given equation, we can simplify it by combining the constants on the left-hand side:

9y - 6 = 16y + 50

Subtracting 9y from both sides gives us:

-6 = 7y + 50

Step 2: Isolate the Variable

Next, we need to isolate the variable (y) on one side of the equation. We can do this by subtracting 50 from both sides:

-6 - 50 = 7y

This simplifies to:

-56 = 7y

Step 3: Solve for the Variable

Finally, we can solve for the variable (y) by dividing both sides of the equation by 7:

y = -56/7

y = -8

Checking the Solution

Once we have solved for the variable, we need to check our solution by plugging it back into the original equation. If the solution is correct, the equation should be true.

Let's plug y = -8 back into the original equation:

9y - 6 = 16y + 50

Substituting y = -8 gives us:

9(-8) - 6 = 16(-8) + 50

-72 - 6 = -128 + 50

-78 = -78

The equation is true, so our solution is correct.

Conclusion

Solving and checking linear equations is an essential skill for students and professionals alike. By following the steps outlined in this article, you can solve linear equations with ease. Remember to simplify the equation, isolate the variable, and check your solution to ensure that it is correct.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid:

Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.
Not checking the solution: Failing to check the solution can lead to incorrect solutions.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

Use inverse operations: Use inverse operations (such as addition and subtraction, multiplication and division) to isolate the variable.
Simplify the equation: Simplify the equation by combining like terms.
Check your solution: Check your solution by plugging it back into the original equation.

Real-World Applications

Linear equations have numerous real-world applications, including:

Physics: Linear equations are used to describe the motion of objects.
Engineering: Linear equations are used to design and optimize systems.
Economics: Linear equations are used to model economic systems.

Conclusion

Introduction

In our previous article, we discussed how to solve and check linear equations. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) 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 simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms on both sides of the equation. This involves adding or subtracting the same value from both sides of the equation.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to use inverse operations to get the variable on one side of the equation. For example, if you have the equation:

2x + 3 = 5

You can subtract 3 from both sides to get:

2x = 2

Then, you can divide both sides by 2 to get:

x = 1

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug the solution back into the original equation and see if it is true. If the solution is correct, the equation should be true.

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

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

Not simplifying the equation
Not isolating the variable
Not checking the solution

Q: How do I use inverse operations to solve linear equations?

A: Inverse operations are used to solve linear equations by getting the variable on one side of the equation. For example, if you have the equation:

2x + 3 = 5

You can subtract 3 from both sides to get:

2x = 2

Then, you can divide both sides by 2 to get:

x = 1

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

A: Linear equations have numerous real-world applications, including:

Physics: Linear equations are used to describe the motion of objects.
Engineering: Linear equations are used to design and optimize systems.
Economics: Linear equations are used to model economic systems.

Q: How do I use linear equations to solve problems in physics?

A: Linear equations are used to describe the motion of objects in physics. For example, if you have a ball rolling down a hill, you can use a linear equation to describe its motion. The equation would be:

s = vt

where s is the distance traveled, v is the velocity, and t is the time.

Q: How do I use linear equations to solve problems in engineering?

A: Linear equations are used to design and optimize systems in engineering. For example, if you have a system with multiple components, you can use a linear equation to describe the behavior of the system. The equation would be:

y = mx + b

where y is the output, m is the slope, x is the input, and b is the intercept.

Q: How do I use linear equations to solve problems in economics?

A: Linear equations are used to model economic systems in economics. For example, if you have a company with multiple products, you can use a linear equation to describe the demand for each product. The equation would be:

p = a + bx

where p is the price, a is the intercept, b is the slope, and x is the quantity demanded.