Solve The Equation:${ 6(t - 1) = 9(t - 4) }$

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, $6(t - 1) = 9(t - 4)$, and provide a step-by-step guide on how to approach it. We will also discuss the importance of linear equations in real-life applications and provide tips on how to simplify and solve them.

What are Linear Equations?

Linear equations are algebraic equations in which the highest power of the variable(s) is 1. They can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation $6(t - 1) = 9(t - 4)$

The given equation is $6(t - 1) = 9(t - 4)$. To solve this equation, we need to isolate the variable $t$. We can start by expanding the parentheses on both sides of the equation.

Expanding the Parentheses

To expand the parentheses, we need to multiply the numbers outside the parentheses by the numbers inside. On the left side of the equation, we have $6(t - 1)$, which can be expanded as $6t - 6$. On the right side of the equation, we have $9(t - 4)$, which can be expanded as $9t - 36$.

The Expanded Equation

After expanding the parentheses, the equation becomes $6t - 6 = 9t - 36$. Now, we can simplify the equation by combining like terms.

Simplifying the Equation

To simplify the equation, we need to combine the like terms on both sides. On the left side, we have $6t$ and $-6$, which can be combined as $6t - 6$. On the right side, we have $9t$ and $-36$, which can be combined as $9t - 36$. The equation now becomes $6t - 6 = 9t - 36$.

Isolating the Variable

To isolate the variable $t$, we need to get all the terms with $t$ on one side of the equation. We can do this by subtracting $6t$ from both sides of the equation.

Subtracting $6t$ from Both Sides

After subtracting $6t$ from both sides, the equation becomes $-6 = 3t - 36$. Now, we can simplify the equation by combining like terms.

Simplifying the Equation

To simplify the equation, we need to combine the like terms on both sides. On the left side, we have $-6$, which remains the same. On the right side, we have $3t$ and $-36$, which can be combined as $3t - 36$. The equation now becomes $-6 = 3t - 36$.

Adding $36$ to Both Sides

To isolate the variable $t$, we need to get all the terms with $t$ on one side of the equation. We can do this by adding $36$ to both sides of the equation.

Adding $36$ to Both Sides

After adding $36$ to both sides, the equation becomes $30 = 3t$. Now, we can simplify the equation by dividing both sides by $3$.

Dividing Both Sides by $3$

After dividing both sides by $3$, the equation becomes $10 = t$. Therefore, the solution to the equation $6(t - 1) = 9(t - 4)$ is $t = 10$.

Conclusion

Solving linear equations is an essential skill for students to master. In this article, we focused on solving the equation $6(t - 1) = 9(t - 4)$ using algebraic manipulation. We expanded the parentheses, simplified the equation, isolated the variable, and finally solved for $t$. We hope that this step-by-step guide has provided a clear understanding of how to approach and solve linear equations.

Tips for Solving Linear Equations

Here are some tips for solving linear equations:

• Read the equation carefully: Before starting to solve the equation, read it carefully to understand what is being asked.
• Use algebraic manipulation: Use algebraic manipulation to simplify the equation and isolate the variable.
• Combine like terms: Combine like terms on both sides of the equation to simplify it.
• Isolate the variable: Isolate the variable by getting all the terms with the variable on one side of the equation.
• Check your solution: Check your solution by plugging it back into the original equation.

Real-Life Applications of Linear Equations

Linear equations have numerous real-life applications, including:

• Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
• Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and computer networks.
• Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and optimization.
• Computer Science: Linear equations are used to solve problems in computer science, including graph theory, network flow, and optimization.

Conclusion

Solving linear equations is an essential skill for students to master. In this article, we focused on solving the equation $6(t - 1) = 9(t - 4)$ using algebraic manipulation. We expanded the parentheses, simplified the equation, isolated the variable, and finally solved for $t$. We hope that this step-by-step guide has provided a clear understanding of how to approach and solve linear equations.

Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand how to approach and solve them. In this article, we will provide a Q&A guide on solving linear equations, including common mistakes to avoid and tips for simplifying and solving them.

Q: What is a linear equation?

A: A linear equation is an algebraic 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 solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by getting all the terms with the variable on one side of the equation. You can do this by using algebraic manipulation, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

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

• Not reading the equation carefully: Before starting to solve the equation, read it carefully to understand what is being asked.
• Not using algebraic manipulation: Use algebraic manipulation to simplify the equation and isolate the variable.
• Not combining like terms: Combine like terms on both sides of the equation to simplify it.
• Not isolating the variable: Isolate the variable by getting all the terms with the variable on one side of the equation.
• Not checking your solution: Check your solution by plugging it back into the original equation.

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 coefficients of the same variable.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an algebraic equation in which the highest power of the variable(s) is 1, while a quadratic equation is an algebraic equation in which the highest power of the variable(s) is 2.

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

A: To solve a system of linear equations, you need to find the values of the variables that satisfy all the equations in the system. You can do this by using substitution or elimination methods.

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

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

• Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
• Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and computer networks.
• Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and optimization.
• Computer Science: Linear equations are used to solve problems in computer science, including graph theory, network flow, and optimization.

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

A: To check your solution to a linear equation, you need to plug it back into the original equation and verify that it is true. If the solution satisfies the equation, then it is the correct solution.

Q: What are some common types of linear equations?

A: Some common types of linear equations include:

• Simple linear equations: These are linear equations in which the coefficient of the variable is 1.
• Linear equations with fractions: These are linear equations in which the coefficients are fractions.
• Linear equations with decimals: These are linear equations in which the coefficients are decimals.
• Linear equations with variables on both sides: These are linear equations in which the variable appears on both sides of the equation.

Conclusion

Solving linear equations is an essential skill for students to master. In this article, we provided a Q&A guide on solving linear equations, including common mistakes to avoid and tips for simplifying and solving them. We hope that this guide has provided a clear understanding of how to approach and solve linear equations.