If $\frac{a}{b} = \frac{3}{4}$ And $a + B = 28$, What Is The Value Of $a$?

Introduction

In this article, we will explore a system of equations involving fractions and linear equations. We will use algebraic techniques to solve for the value of $a$ in the given equation $\frac{a}{b} = \frac{3}{4}$ and $a + b = 28$. This problem requires a combination of fraction manipulation and linear equation solving skills.

Understanding the Problem

We are given two equations:

1. $\frac{a}{b} = \frac{3}{4}$
2. $a + b = 28$

Our goal is to find the value of $a$ that satisfies both equations.

Step 1: Manipulating the Fraction Equation

To start solving the problem, we can manipulate the fraction equation to isolate $a$. We can do this by cross-multiplying the fractions:

$$\frac{a}{b} = \frac{3}{4} \Rightarrow 4a = 3b$$

This equation can be rewritten as:

$$a = \frac{3}{4}b$$

Step 2: Substituting into the Linear Equation

Now that we have an expression for $a$ in terms of $b$, we can substitute this expression into the linear equation:

$$a + b = 28 \Rightarrow \frac{3}{4}b + b = 28$$

Step 3: Combining Like Terms

We can combine the like terms on the left-hand side of the equation:

$$\frac{3}{4}b + b = \frac{7}{4}b$$

So, the equation becomes:

$$\frac{7}{4}b = 28$$

Step 4: Solving for $b$

To solve for $b$, we can multiply both sides of the equation by the reciprocal of $\frac{7}{4}$, which is $\frac{4}{7}$:

$$\frac{4}{7} \cdot \frac{7}{4}b = \frac{4}{7} \cdot 28$$

This simplifies to:

$$b = 16$$

Step 5: Finding the Value of $a$

Now that we have the value of $b$, we can substitute it back into the expression for $a$:

$$a = \frac{3}{4}b \Rightarrow a = \frac{3}{4} \cdot 16$$

This simplifies to:

$$a = 12$$

Conclusion

In this article, we used algebraic techniques to solve a system of equations involving fractions and linear equations. We manipulated the fraction equation to isolate $a$, substituted into the linear equation, combined like terms, solved for $b$, and finally found the value of $a$. The final answer is $\boxed{12}$.

Additional Tips and Variations

• To check the solution, we can substitute the values of $a$ and $b$ back into the original equations to ensure that they are satisfied.
• We can also use other algebraic techniques, such as substitution or elimination, to solve the system of equations.
• If the problem had multiple solutions, we would need to consider all possible values of $a$ and $b$ that satisfy the equations.

Real-World Applications

This problem has real-world applications in various fields, such as:

• Finance: When calculating interest rates or investment returns, we may encounter systems of equations involving fractions and linear equations.
• Science: In physics and engineering, we often use systems of equations to model real-world phenomena, such as motion or electrical circuits.
• Business: In management and economics, we may use systems of equations to optimize business decisions or model market trends.

Final Thoughts

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Q: What is a system of equations?

A system of equations is a set of two or more equations that involve the same variables. In the case of the problem we solved earlier, we had two equations:

1. $\frac{a}{b} = \frac{3}{4}$
2. $a + b = 28$

Q: Why do we need to solve systems of equations?

Solving systems of equations is essential in various fields, such as finance, science, and business. It helps us to:

• Model real-world phenomena
• Make informed decisions
• Optimize business outcomes
• Understand complex relationships between variables

Q: What are some common techniques for solving systems of equations?

Some common techniques for solving systems of equations include:

• Substitution method
• Elimination method
• Graphical method
• Algebraic method (as we used in the problem we solved earlier)

Q: How do I choose the right technique for solving a system of equations?

The choice of technique depends on the type of equations and the variables involved. For example:

• If the equations are linear, we can use the substitution or elimination method.
• If the equations are non-linear, we may need to use graphical or algebraic methods.
• If the equations involve fractions, we may need to use algebraic methods to manipulate the fractions.

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

Some common mistakes to avoid when solving systems of equations include:

• Not checking the validity of the solutions
• Not considering all possible solutions
• Not using the correct technique for the type of equations
• Not simplifying the equations before solving

Q: How do I check the validity of the solutions?

To check the validity of the solutions, we can:

• Substitute the values of the variables back into the original equations
• Check if the solutions satisfy all the equations
• Use graphical or algebraic methods to verify the solutions

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

Solving systems of equations has many real-world applications, including:

• Finance: Calculating interest rates, investment returns, and risk management
• Science: Modeling motion, electrical circuits, and chemical reactions
• Business: Optimizing business decisions, modeling market trends, and predicting sales
• Engineering: Designing bridges, buildings, and other structures

Q: Can I use technology to solve systems of equations?

Yes, we can use technology, such as calculators, computers, or software, to solve systems of equations. This can be especially helpful for complex systems or when we need to visualize the solutions.

Q: What are some tips for solving systems of equations?

Some tips for solving systems of equations include:

• Read the problem carefully and understand what is being asked
• Choose the right technique for the type of equations
• Simplify the equations before solving
• Check the validity of the solutions
• Use technology to verify the solutions

Conclusion

Solving systems of equations is an essential skill in various fields, and it requires a combination of algebraic techniques and problem-solving skills. By following the steps outlined in this article and avoiding common mistakes, we can find the solutions to complex systems of equations.