Solving Equations: Find X = -1/4 Solutions

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Hey guys! Today, we're diving into some equations to figure out which ones have x=−14x = -\frac{1}{4} as the solution. It's like a little math puzzle, and we're going to solve it together step by step. So, grab your pencils, and let's get started!

Checking the Equations

We need to substitute x=−14x = -\frac{1}{4} into each equation and see if the equation holds true. This means the left side of the equation should be equal to the right side after the substitution. Let's go through each equation one by one.

Equation 1: x−14=−12x - \frac{1}{4} = -\frac{1}{2}

First, let's substitute x=−14x = -\frac{1}{4} into the equation:

−14−14=−12- \frac{1}{4} - \frac{1}{4} = -\frac{1}{2}

Now, let's simplify the left side:

−14−14=−24- \frac{1}{4} - \frac{1}{4} = -\frac{2}{4}

Reduce the fraction:

−24=−12- \frac{2}{4} = -\frac{1}{2}

So, we have:

−12=−12- \frac{1}{2} = -\frac{1}{2}

This equation is true! Therefore, x=−14x = -\frac{1}{4} is a solution for the equation x−14=−12x - \frac{1}{4} = -\frac{1}{2}.

Why is this important? Understanding how to substitute values into equations is crucial in algebra. It helps us verify solutions and solve for unknowns. In this case, we're verifying whether a given value is indeed a solution to the equation. This skill forms the foundation for solving more complex algebraic problems, such as systems of equations and inequalities. It also reinforces the concept of equality—both sides of an equation must remain balanced. For example, if we add or subtract a number from one side, we must do the same to the other side to maintain the equality. Moreover, by mastering these basics, students gain confidence in their ability to manipulate equations and solve real-world problems involving mathematical modeling. This is a fundamental skill that will assist them throughout their academic and professional lives.

Equation 2: x−12=14x - \frac{1}{2} = \frac{1}{4}

Substitute x=−14x = -\frac{1}{4} into the equation:

−14−12=14- \frac{1}{4} - \frac{1}{2} = \frac{1}{4}

To combine the fractions on the left side, we need a common denominator, which is 4:

−14−24=14- \frac{1}{4} - \frac{2}{4} = \frac{1}{4}

Now, let's simplify the left side:

−34=14- \frac{3}{4} = \frac{1}{4}

This equation is not true, since −34- \frac{3}{4} is not equal to 14\frac{1}{4}. Therefore, x=−14x = -\frac{1}{4} is not a solution for the equation x−12=14x - \frac{1}{2} = \frac{1}{4}.

Diving deeper into this topic, we can see that fractions play a crucial role in various areas of mathematics and real-life applications. Fractions are not just abstract numbers; they represent parts of a whole and enable us to express quantities that are not whole numbers. From cooking and baking, where we measure ingredients using fractions like 12\frac{1}{2} cup or 14\frac{1}{4} teaspoon, to construction and engineering, where precise measurements are essential, fractions are indispensable. In finance, fractions appear in interest rates, stock prices, and percentage calculations. Therefore, being comfortable with fraction arithmetic—adding, subtracting, multiplying, and dividing fractions—is essential for everyday problem-solving. Furthermore, the ability to convert fractions to decimals and percentages allows for easier comparison and computation, particularly when dealing with electronic calculators or computer software. For students, mastering fractions builds a solid foundation for more advanced math concepts such as algebra, geometry, and calculus, where fractions are used extensively.

Equation 3: x−12=−34x - \frac{1}{2} = -\frac{3}{4}

Substitute x=−14x = -\frac{1}{4} into the equation:

−14−12=−34- \frac{1}{4} - \frac{1}{2} = -\frac{3}{4}

Again, we need a common denominator to combine the fractions on the left side. The common denominator is 4:

−14−24=−34- \frac{1}{4} - \frac{2}{4} = -\frac{3}{4}

Now, let's simplify the left side:

−34=−34- \frac{3}{4} = -\frac{3}{4}

This equation is true! So, x=−14x = -\frac{1}{4} is a solution for the equation x−12=−34x - \frac{1}{2} = -\frac{3}{4}.

Understanding Equality in Equations: Equations are mathematical statements that assert the equality of two expressions. The fundamental principle when working with equations is that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the equality. This principle is rooted in the properties of equality, such as the addition, subtraction, multiplication, and division properties. For instance, if a=ba = b, then a+c=b+ca + c = b + c for any number cc. This ensures that the equation remains balanced. When solving equations, we use these properties to isolate the variable on one side of the equation, thereby finding its value. In our case, substituting x=−14x = -\frac{1}{4} into the equation and showing that both sides are equal confirms that x=−14x = -\frac{1}{4} is indeed a solution. This process of verifying solutions reinforces the concept of equality and the importance of maintaining balance in equations.

Equation 4: 12+x=14\frac{1}{2} + x = \frac{1}{4}

Substitute x=−14x = -\frac{1}{4} into the equation:

12+(−14)=14\frac{1}{2} + (-\frac{1}{4}) = \frac{1}{4}

Simplify the left side. The common denominator is 4:

24−14=14\frac{2}{4} - \frac{1}{4} = \frac{1}{4}

Now, simplify the left side:

14=14\frac{1}{4} = \frac{1}{4}

This equation is true! So, x=−14x = -\frac{1}{4} is a solution for the equation 12+x=14\frac{1}{2} + x = \frac{1}{4}.

Additive Inverses and Zero are very important in simplifying expressions. An additive inverse of a number is the number that, when added to the original number, results in zero. For example, the additive inverse of 5 is -5, and the additive inverse of -3 is 3. In equations, we often use additive inverses to isolate variables and simplify expressions. For instance, in the equation x+5=8x + 5 = 8, we can add the additive inverse of 5 (which is -5) to both sides of the equation to isolate xx: x+5+(−5)=8+(−5)x + 5 + (-5) = 8 + (-5), which simplifies to x=3x = 3. Understanding additive inverses is crucial for solving equations and understanding more complex mathematical concepts. Moreover, it reinforces the concept of balance in equations, emphasizing that whatever operation we perform on one side, we must also perform on the other side to maintain the equality. It is a simple concept that will assist students to solve the equation and simplify the math expression.

Equation 5: x+12=−14x + \frac{1}{2} = -\frac{1}{4}

Substitute x=−14x = -\frac{1}{4} into the equation:

−14+12=−14- \frac{1}{4} + \frac{1}{2} = -\frac{1}{4}

Simplify the left side. The common denominator is 4:

−14+24=−14- \frac{1}{4} + \frac{2}{4} = -\frac{1}{4}

Now, simplify the left side:

14=−14\frac{1}{4} = -\frac{1}{4}

This equation is not true because 14\frac{1}{4} is not equal to −14- \frac{1}{4}. Therefore, x=−14x = -\frac{1}{4} is not a solution for the equation x+12=−14x + \frac{1}{2} = -\frac{1}{4}.

Conclusion

Alright, guys! We've checked all the equations and found that x=−14x = -\frac{1}{4} is a solution for the following equations:

  • x−14=−12x - \frac{1}{4} = -\frac{1}{2}
  • x−12=−34x - \frac{1}{2} = -\frac{3}{4}
  • 12+x=14\frac{1}{2} + x = \frac{1}{4}

I hope this was helpful and you now have a better understanding of how to verify solutions to equations! Keep practicing, and you'll become math pros in no time! If you have any questions, feel free to ask. Happy solving!