If $3x + 4y = 7$, Which Of The Following Expressions Is Equivalent To $y$?Choose 1 Answer:A. $-\frac{3}{4}x + \frac{7}{4}$B. $-3x + 7$C. $\frac{3}{4}x + \frac{7}{4}$D. $3x + 7$

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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 type of linear equation, namely the equation 3x+4y=73x + 4y = 7. Our goal is to find an expression equivalent to yy, which is a key concept in algebra.

Understanding the Equation

The given equation is 3x+4y=73x + 4y = 7. To solve for yy, we need to isolate the variable yy on one side of the equation. This can be done by using algebraic operations such as addition, subtraction, multiplication, and division.

Step 1: Subtract 3x from Both Sides

To isolate yy, we first need to get rid of the term 3x3x on the left-hand side of the equation. We can do this by subtracting 3x3x from both sides of the equation.

3x + 4y = 7
-3x -3x
4y = 7 - 3x

Step 2: Simplify the Right-Hand Side

Now that we have 4y4y on the left-hand side, we need to simplify the right-hand side of the equation. We can do this by subtracting 3x3x from 77.

4y = 7 - 3x
4y = 4 - 3x

Step 3: Divide Both Sides by 4

To isolate yy, we need to get rid of the coefficient 44 on the left-hand side of the equation. We can do this by dividing both sides of the equation by 44.

4y = 4 - 3x
\frac{4y}{4} = \frac{4 - 3x}{4}
y = \frac{4 - 3x}{4}

Step 4: Simplify the Right-Hand Side

Now that we have yy on the left-hand side, we need to simplify the right-hand side of the equation. We can do this by dividing the numerator and denominator by their greatest common divisor, which is 44.

y = \frac{4 - 3x}{4}
y = \frac{1 - \frac{3}{4}x}{1}
y = 1 - \frac{3}{4}x

Conclusion

In conclusion, the expression equivalent to yy is 1−34x1 - \frac{3}{4}x. This can be rewritten as −34x+1-\frac{3}{4}x + 1, which is equivalent to −34x+74-\frac{3}{4}x + \frac{7}{4}.

Answer

The correct answer is:

  • A. −34x+74-\frac{3}{4}x + \frac{7}{4}

Discussion

This problem requires students to apply algebraic operations to solve for yy. The key concept here is to isolate the variable yy on one side of the equation. Students need to understand the order of operations and apply the correct algebraic operations to simplify the equation.

Tips and Variations

  • To make this problem more challenging, students can be asked to solve for xx instead of yy.
  • Students can also be asked to solve a system of linear equations, where the given equation is one of the equations in the system.
  • To make this problem more accessible, students can be given a simpler equation to solve, such as 2x+3y=52x + 3y = 5.

Real-World Applications

Solving linear equations is a crucial skill in many real-world applications, such as:

  • Finance: Solving linear equations is used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Solving linear equations is used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Solving linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

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 provide a Q&A guide to help students understand and solve 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 + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable(s) on one side of the equation. You can do this by using algebraic operations such as addition, subtraction, multiplication, and division.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Subtract or add the same value to both sides: To isolate the variable(s), you need to get rid of the constant term on the same side as the variable(s).
  2. Multiply or divide both sides by the same value: To isolate the variable(s), you need to get rid of the coefficient on the same side as the variable(s).
  3. Simplify the equation: Once you have isolated the variable(s), you need to simplify the equation by combining like terms.

Q: How do I solve for x in a linear equation?

A: To solve for x in a linear equation, you need to isolate x on one side of the equation. You can do this by using algebraic operations such as addition, subtraction, multiplication, and division.

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(s) is 1, while a quadratic equation is an 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-world applications of linear equations?

A: Linear equations have many real-world applications, such as:

  • Finance: Solving linear equations is used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Solving linear equations is used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Solving linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

  • Solving problems: Try solving linear equations on your own or with the help of a tutor or online resource.
  • Using online resources: There are many online resources available that provide practice problems and exercises to help you improve your skills.
  • Taking online courses: You can take online courses or watch video tutorials to learn more about solving linear equations.

Conclusion

In conclusion, solving linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the steps to solve a linear equation and practicing regularly, you can improve your skills and become proficient in solving linear equations.