Solve For $q$.${ 3 - Q = \frac{3}{4} + \frac{3}{2}q }$q = \square$

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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, where the variable q is isolated on one side of the equation. We will use a step-by-step approach to solve the equation and provide a clear explanation of each step.

The Equation

The given equation is:

3−q=34+32q3 - q = \frac{3}{4} + \frac{3}{2}q

Our goal is to isolate the variable q on one side of the equation.

Step 1: Multiply Both Sides by 4

To eliminate the fractions, we will multiply both sides of the equation by 4. This will give us:

12−4q=3+6q12 - 4q = 3 + 6q

Step 2: Add 4q to Both Sides

Next, we will add 4q to both sides of the equation to get:

12=3+10q12 = 3 + 10q

Step 3: Subtract 3 from Both Sides

Now, we will subtract 3 from both sides of the equation to get:

9=10q9 = 10q

Step 4: Divide Both Sides by 10

Finally, we will divide both sides of the equation by 10 to isolate the variable q:

q=910q = \frac{9}{10}

Conclusion

In this article, we have solved a linear equation by isolating the variable q on one side of the equation. We have used a step-by-step approach to eliminate fractions, add and subtract terms, and finally divide both sides by 10 to get the value of q. This type of problem is a great example of how to solve linear equations and is a fundamental concept in mathematics.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
  • When multiplying or dividing both sides of an equation by a term, make sure to multiply or divide both sides by the same term.
  • When adding or subtracting terms, make sure to add or subtract the same terms on both sides of the equation.

Real-World Applications

Linear equations have many real-world 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, inflation, and unemployment.

Practice Problems

Here are a few practice problems to help you reinforce your understanding of solving linear equations:

  1. Solve the equation: 2x+3=5x−22x + 3 = 5x - 2
  2. Solve the equation: x−2=12x+1x - 2 = \frac{1}{2}x + 1
  3. Solve the equation: 3y+2=2y−13y + 2 = 2y - 1

Solutions

  1. x=73x = \frac{7}{3}
  2. x=33=1x = \frac{3}{3} = 1
  3. y=−12y = -\frac{1}{2}

Conclusion

Introduction

In our previous article, we solved a linear equation by isolating the variable q on one side of the equation. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to 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. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants.

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

A: The steps to solve a linear equation are:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding or subtracting the same term to both sides of the equation.
  3. Multiply or divide both sides of the equation by the same term to eliminate fractions.
  4. Add or subtract the same term to both sides of the equation to isolate the variable.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you can combine like terms by adding or subtracting the same term to both sides of the equation. For example, if you have the equation 2x + 3 = 5x - 2, you can simplify it by combining the like terms:

2x + 3 = 5x - 2

Subtract 2x from both sides:

3 = 3x - 2

Add 2 to both sides:

5 = 3x

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

A: To isolate the variable in a linear equation, you can add or subtract the same term to both sides of the equation. For example, if you have the equation x - 2 = 3, you can isolate the variable by adding 2 to both sides:

x - 2 + 2 = 3 + 2

x = 5

Q: How do I eliminate fractions in a linear equation?

A: To eliminate fractions in a linear equation, you can multiply or divide both sides of the equation by the same term. For example, if you have the equation 1/2x + 3 = 5, you can eliminate the fraction by multiplying both sides by 2:

(1/2)x + 3 = 5

Multiply both sides by 2:

x + 6 = 10

Subtract 6 from both sides:

x = 4

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 before solving it.
  • Not isolating the variable on one side of the equation.
  • Not eliminating fractions before solving the equation.
  • Not checking the solution to make sure it is correct.

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

A: To check your solution to a linear equation, you can plug the solution back into the original equation and make sure it is true. For example, if you have the equation x - 2 = 3 and you solve it to get x = 5, you can plug x = 5 back into the original equation to check:

5 - 2 = 3

3 = 3

Since the equation is true, the solution x = 5 is correct.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and techniques used to solve linear equations. We have covered topics such as simplifying linear equations, isolating the variable, eliminating fractions, and checking solutions. By following these steps and avoiding common mistakes, you can become proficient in solving linear equations and apply this skill to a wide range of problems in mathematics and other fields.