Determine The Integer To Make The Following Statement True:$\[ 12 - \Delta = 22 \\]A. 10 B. 12 C. \[$-10\$\] D. No Solution.

Introduction

Mathematical equations are a fundamental part of mathematics, and solving for unknown variables is a crucial skill that every student should possess. In this article, we will focus on solving for an unknown integer in a simple mathematical equation. We will use the given equation $12 - \Delta = 22$ to determine the value of the unknown integer $\Delta$. Our goal is to find the correct answer among the options provided.

Understanding the Equation

The given equation is $12 - \Delta = 22$. To solve for $\Delta$, we need to isolate the variable on one side of the equation. We can do this by adding $\Delta$ to both sides of the equation, which will cancel out the negative sign and leave us with the value of $\Delta$.

Step 1: Add $\Delta$ to Both Sides of the Equation

When we add $\Delta$ to both sides of the equation, we get:

$$12 - \Delta + \Delta = 22 + \Delta$$

The $\Delta$ terms cancel out, leaving us with:

$$12 = 22 + \Delta$$

Step 2: Subtract 22 from Both Sides of the Equation

To isolate $\Delta$, we need to get rid of the 22 on the right-hand side of the equation. We can do this by subtracting 22 from both sides of the equation:

$$12 - 22 = 22 + \Delta - 22$$

This simplifies to:

$$-10 = \Delta$$

Conclusion

Based on our calculations, we have found that the value of $\Delta$ is $-10$. This means that the correct answer is:

C. $-10$

Why is this the Correct Answer?

To verify that our answer is correct, let's plug in the value of $\Delta$ back into the original equation:

$$12 - (-10) = 22$$

Simplifying this expression, we get:

$$12 + 10 = 22$$

Which is indeed true. Therefore, we can conclude that the correct answer is indeed $-10$.

What if there was no solution?

In some cases, it may be possible that there is no solution to the equation. This can happen when the equation is inconsistent, meaning that it is impossible to satisfy the equation with any value of the variable. In such cases, the correct answer would be:

D. No solution

However, in this case, we have found a clear solution to the equation, which is $-10$.

Conclusion

$$$$$$$$

Introduction

In our previous article, we solved for the unknown integer $\Delta$ in the equation $12 - \Delta = 22$. We found that the value of $\Delta$ is $-10$, which is the correct answer among the options provided. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the difference between an equation and an expression?

A: An equation is a statement that says two mathematical expressions are equal, whereas an expression is a group of numbers, variables, and mathematical operations. For example, $2x + 3$ is an expression, while $2x + 3 = 5$ is an equation.

Q: How do I know if an equation is true or false?

A: To determine if an equation is true or false, you need to check if the equation is consistent. If the equation is consistent, it means that there is a solution to the equation, and you can find the value of the variable. If the equation is inconsistent, it means that there is no solution to the equation, and it is false.

Q: What is the order of operations in solving an equation?

A: The order of operations in solving an equation is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I isolate a variable in an equation?

A: To isolate a variable in an equation, you need to get the variable by itself on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same non-zero value.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, $2x + 3 = 5$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the significance of the variable in an equation?

A: The variable in an equation represents a value that is unknown or changing. The variable is often represented by a letter, such as $x$ or $y$. The variable is used to solve for the value of the unknown quantity.

Conclusion

In conclusion, we have provided a Q&A section to help clarify any doubts or questions that readers may have about solving for the unknown integer in a mathematical equation. We have discussed the difference between an equation and an expression, the order of operations in solving an equation, and how to isolate a variable in an equation. We have also discussed the difference between a linear equation and a quadratic equation, and how to solve a quadratic equation using the quadratic formula.