Which Of These Equations, When Solved, Gives A Different Value Of $x$ Than The Other Three?A. 9.1 = − 0.2 X + 10 9.1 = -0.2x + 10 9.1 = − 0.2 X + 10 B. 10 = 9.1 + 0.2 X 10 = 9.1 + 0.2x 10 = 9.1 + 0.2 X C. 10 − 0.2 X = 9.1 10 - 0.2x = 9.1 10 − 0.2 X = 9.1 D. 9.1 − 10 = 0.2 X 9.1 - 10 = 0.2x 9.1 − 10 = 0.2 X

When it comes to solving equations, it's essential to understand the underlying principles and how they apply to different scenarios. In this article, we'll delve into a set of equations and explore which one yields a distinct value of x when solved.

Understanding the Basics of Equations

Equations are mathematical statements that express the equality of two expressions. They can be linear or non-linear, and solving them involves finding the value of the variable(s) that makes the equation true. In the context of this discussion, we'll focus on linear equations, which are equations in which the highest power of the variable(s) is 1.

The Four Equations

We have four equations to consider:

A. $9.1 = -0.2x + 10$ B. $10 = 9.1 + 0.2x$ C. $10 - 0.2x = 9.1$ D. $9.1 - 10 = 0.2x$

At first glance, these equations may seem similar, but they differ in their structure and the operations involved. Let's take a closer look at each equation and see how they can be solved.

Equation A: $9.1 = -0.2x + 10$

To solve this equation, we need to isolate the variable x. We can start by subtracting 10 from both sides of the equation:

$9.1 - 10 = -0.2x + 10 - 10$

This simplifies to:

$-0.9 = -0.2x$

Next, we can multiply both sides of the equation by -5 to eliminate the fraction:

$-0.9 \times -5 = -0.2x \times -5$

This gives us:

$4.5 = x$

So, the value of x in equation A is 4.5.

Equation B: $10 = 9.1 + 0.2x$

To solve this equation, we can start by subtracting 9.1 from both sides of the equation:

$10 - 9.1 = 0.2x$

This simplifies to:

$0.9 = 0.2x$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$0.9 \times 5 = 0.2x \times 5$

This gives us:

$4.5 = x$

So, the value of x in equation B is also 4.5.

Equation C: $10 - 0.2x = 9.1$

To solve this equation, we can start by subtracting 9.1 from both sides of the equation:

$10 - 9.1 - 0.2x = 9.1 - 9.1$

This simplifies to:

$0.9 - 0.2x = 0$

Next, we can add 0.2x to both sides of the equation:

$0.9 = 0.2x$

This is the same as equation B, and we know that the value of x in equation B is 4.5.

Equation D: $9.1 - 10 = 0.2x$

To solve this equation, we can start by subtracting 9.1 from both sides of the equation:

$-0.9 = 0.2x$

Next, we can multiply both sides of the equation by -5 to eliminate the fraction:

$-0.9 \times -5 = 0.2x \times -5$

This gives us:

$4.5 = x$

So, the value of x in equation D is also 4.5.

The Conclusion

At first glance, it may seem like all four equations yield the same value of x, which is 4.5. However, upon closer inspection, we can see that equation D is actually a rearrangement of equation A, and equation C is a rearrangement of equation B. This means that equations A, B, and D are essentially the same, and they all yield the same value of x.

In contrast, equation C is a distinct equation that yields a different value of x. To see this, we can solve equation C by adding 0.2x to both sides of the equation:

$10 - 0.2x + 0.2x = 9.1 + 0.2x$

This simplifies to:

$10 = 9.1 + 0.2x$

This is the same as equation B, but with the opposite sign. To solve for x, we can subtract 9.1 from both sides of the equation:

$10 - 9.1 = 0.2x$

This gives us:

$0.9 = 0.2x$

Next, we can multiply both sides of the equation by -5 to eliminate the fraction:

$-0.9 = 0.2x \times -5$

This gives us:

$-4.5 = x$

So, the value of x in equation C is actually -4.5, which is different from the value of x in equations A, B, and D.

The Final Answer

In our previous article, we explored a set of equations and identified which one yields a distinct value of x when solved. In this article, we'll answer some of the most frequently asked questions related to solving equations.

Q: What is the difference between a linear equation and a non-linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. For example, the equation 2x + 3 = 5 is a linear equation. A non-linear equation, on the other hand, is an equation in which the highest power of the variable(s) is greater than 1. For example, the equation x^2 + 2x + 1 = 0 is a non-linear equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

1. Simplify the equation by combining like terms.
2. Isolate the variable by adding or subtracting the same value to both sides of the equation.
3. Multiply or divide both sides of the equation by the same value to eliminate any fractions.
4. Check your solution by plugging it back into the original equation.

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

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

1. Parentheses: Evaluate any 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 handle fractions when solving an equation?

A: When solving an equation with fractions, you can multiply or divide both sides of the equation by the same value to eliminate any fractions. For example, if you have the equation 1/2x = 3, you can multiply both sides of the equation by 2 to get x = 6.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously. For example, the system of equations:

x + y = 3 2x - y = 1

is a system of two equations. A single equation, on the other hand, is a single equation that is solved independently. For example, the equation x + 2 = 5 is a single equation.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the following methods:

1. Substitution: Substitute the expression for one variable from one equation into the other equation.
2. Elimination: Add or subtract the two equations to eliminate one variable.
3. Graphing: Graph the two equations on a coordinate plane and find the point of intersection.

Q: What is the importance of checking your solution when solving an equation?

A: Checking your solution is an essential step when solving an equation. It ensures that your solution is correct and that you have not made any mistakes. If you plug your solution back into the original equation and it does not satisfy the equation, then you know that your solution is incorrect.

Conclusion

Solving equations is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. By following the steps outlined in this article, you can solve equations with confidence and accuracy. Remember to always check your solution and to use the order of operations when solving an equation.