Complete The Equation With Numbers That Make The Expression On The Right Side Of The Equal Sign Equivalent To The Expression On The Left.$75a + 25b = 75$

Introduction

In mathematics, equations are a fundamental concept that help us solve problems and understand relationships between variables. In this article, we will focus on solving a simple equation with two variables, $a$ and $b$. The equation is given as $75a + 25b = 75$, and our goal is to find the values of $a$ and $b$ that make the expression on the right side of the equal sign equivalent to the expression on the left.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at what it means. The equation is a linear equation, which means it can be represented as a straight line on a graph. The equation has two variables, $a$ and $b$, and the coefficients of these variables are $75$ and $25$, respectively. The constant term on the right side of the equation is $75$.

Solving for $a$ and $b$

To solve for $a$ and $b$, we need to isolate one of the variables. Let's start by isolating $a$. We can do this by subtracting $25b$ from both sides of the equation:

$$75a + 25b - 25b = 75 - 25b$$

This simplifies to:

$$75a = 75 - 25b$$

Now, we can divide both sides of the equation by $75$ to solve for $a$:

$$a = \frac{75 - 25b}{75}$$

Simplifying the Expression

We can simplify the expression for $a$ by dividing the numerator and denominator by their greatest common divisor, which is $25$:

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

Finding the Values of $a$ and $b$

Now that we have the expression for $a$, we can find the values of $a$ and $b$ that satisfy the equation. We can do this by substituting different values of $b$ into the expression for $a$ and solving for $a$.

Let's start by substituting $b = 0$ into the expression for $a$:

$$a = \frac{3 - 0}{3}$$

This simplifies to:

$$a = 1$$

Now, we can substitute $a = 1$ into the original equation to find the value of $b$:

$$75(1) + 25b = 75$$

This simplifies to:

$$75 + 25b = 75$$

Subtracting $75$ from both sides of the equation gives us:

$$25b = 0$$

Dividing both sides of the equation by $25$ gives us:

$$b = 0$$

Conclusion

In this article, we solved the equation $75a + 25b = 75$ by isolating one of the variables and simplifying the expression. We found that the values of $a$ and $b$ that satisfy the equation are $a = 1$ and $b = 0$. This is just one possible solution to the equation, and there may be other values of $a$ and $b$ that also satisfy the equation.

Tips and Tricks

• When solving equations with two variables, it's often helpful to isolate one of the variables first.
• Simplifying expressions can make it easier to solve for the variables.
• Substituting different values of one variable into the expression for the other variable can help you find the values of both variables.

Common Mistakes

• Failing to isolate one of the variables can make it difficult to solve for the other variable.
• Not simplifying expressions can make it harder to solve for the variables.
• Not checking your work can lead to incorrect solutions.

Real-World Applications

Solving equations with two variables has many real-world applications, such as:

• Finance: Solving equations with two variables can help you understand the relationships between different financial variables, such as interest rates and investment returns.
• Science: Solving equations with two variables can help you understand the relationships between different scientific variables, such as temperature and pressure.
• Engineering: Solving equations with two variables can help you understand the relationships between different engineering variables, such as stress and strain.

Conclusion

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Introduction

In our previous article, we solved the equation $75a + 25b = 75$ by isolating one of the variables and simplifying the expression. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the equation $75a + 25b = 75$ trying to tell us?

A: The equation $75a + 25b = 75$ is trying to tell us that the sum of $75a$ and $25b$ is equal to $75$. In other words, the equation is saying that the value of $75a + 25b$ is $75$.

Q: How do we solve for $a$ and $b$ in the equation $75a + 25b = 75$?

A: To solve for $a$ and $b$, we need to isolate one of the variables. We can do this by subtracting $25b$ from both sides of the equation:

$$75a + 25b - 25b = 75 - 25b$$

This simplifies to:

$$75a = 75 - 25b$$

Now, we can divide both sides of the equation by $75$ to solve for $a$:

$$a = \frac{75 - 25b}{75}$$

Q: How do we simplify the expression for $a$?

A: We can simplify the expression for $a$ by dividing the numerator and denominator by their greatest common divisor, which is $25$:

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

Q: What are the values of $a$ and $b$ that satisfy the equation $75a + 25b = 75$?

A: The values of $a$ and $b$ that satisfy the equation $75a + 25b = 75$ are $a = 1$ and $b = 0$.

Q: How do we find the values of $a$ and $b$ that satisfy the equation $75a + 25b = 75$?

A: We can find the values of $a$ and $b$ that satisfy the equation $75a + 25b = 75$ by substituting different values of $b$ into the expression for $a$ and solving for $a$.

Q: What are some common mistakes to avoid when solving the equation $75a + 25b = 75$?

A: Some common mistakes to avoid when solving the equation $75a + 25b = 75$ include:

• Failing to isolate one of the variables
• Not simplifying expressions
• Not checking your work

Q: What are some real-world applications of solving the equation $75a + 25b = 75$?

A: Some real-world applications of solving the equation $75a + 25b = 75$ include:

• Finance: Solving equations with two variables can help you understand the relationships between different financial variables, such as interest rates and investment returns.
• Science: Solving equations with two variables can help you understand the relationships between different scientific variables, such as temperature and pressure.
• Engineering: Solving equations with two variables can help you understand the relationships between different engineering variables, such as stress and strain.

Conclusion

In conclusion, solving the equation $75a + 25b = 75$ requires a step-by-step approach, including isolating one of the variables, simplifying the expression, and substituting different values of one variable into the expression for the other variable. By following these steps, you can find the values of $a$ and $b$ that satisfy the equation. We hope this Q&A guide has helped you understand the solution and answer any questions you may have.