Solving Equations: Step-by-Step Solutions

Nov 4, 2025 by SLV Team 42 views

Hey guys! Let's dive into the world of solving equations. This guide will walk you through various types of equations, providing clear explanations and step-by-step solutions. We'll cover everything from simple quadratic equations to systems of equations. Ready to sharpen those math skills? Let's get started!

A) Solving the Quadratic Equation: X² - 3x = 0

Alright, let's tackle our first equation: X² - 3x = 0. This is a quadratic equation, which means it has a term with x raised to the power of 2. One of the easiest ways to solve this is by factoring. Factoring involves finding expressions that, when multiplied together, give you the original equation. In this case, we can see that both terms on the left side (X² and -3x) have a common factor: x. To factor, we pull out the common factor x: x(x - 3) = 0. Now, for this product to equal zero, either the first term (x) must equal zero, or the second term (x - 3) must equal zero. So, we have two possible solutions, let's solve for each one.

Solution 1: x = 0
Solution 2: x - 3 = 0. Adding 3 to both sides gives us x = 3.

So, the solutions to the equation X² - 3x = 0 are x = 0 and x = 3. These are the values of x that, when plugged back into the original equation, make the equation true. See? Not so bad, right? We simply factored the equation, identified the values, and that is all!

This method of factoring is often the quickest way to solve a quadratic equation, provided it's easily factorable. Other methods, such as using the quadratic formula, can be used for more complicated equations, but we'll get to those later. The key is recognizing the structure of the equation and choosing the most efficient method for finding the values. Remember, practice makes perfect! The more you work through these problems, the more familiar you'll become with the patterns and tricks involved.

Understanding Quadratic Equations

Understanding quadratic equations is crucial in various fields, from physics and engineering to economics and computer science. These equations describe parabolic curves, which are fundamental in understanding projectile motion, optimizing designs, and modeling growth and decay. In our case, the equation X² - 3x = 0 represents a parabola that intersects the x-axis at the points x = 0 and x = 3. The process of finding these points (the solutions) allows us to analyze and predict the behavior of the system the equation represents. It's like having a key to unlock the underlying dynamics of many real-world phenomena.

B) Solving the Quadratic Equation: 5x² - 7x + 15 = 0

Okay, let's move on to the next equation: 5x² - 7x + 15 = 0. This is another quadratic equation, but this time, it might not be as easy to factor as the previous one. When factoring isn't straightforward, the quadratic formula is your best friend. The quadratic formula is a universal method that works for any quadratic equation in the form ax² + bx + c = 0. The formula is: x = (-b ± √(b² - 4ac)) / 2a.

Let's apply this to our equation: 5x² - 7x + 15 = 0.

Here, a = 5, b = -7, and c = 15.

Let's substitute these values into the formula:

x = ( -(-7) ± √((-7)² - 4 * 5 * 15)) / (2 * 5)
x = (7 ± √(49 - 300)) / 10
x = (7 ± √(-251)) / 10

Notice that we have a negative number inside the square root. This means the solutions will involve imaginary numbers. The square root of -1 is represented by i, so: x = (7 ± i√251) / 10.

Therefore, the solutions to the equation 5x² - 7x + 15 = 0 are x = (7 + i√251) / 10 and x = (7 - i√251) / 10. These are complex solutions, meaning they include an imaginary part. This indicates that the parabola represented by this equation doesn't intersect the x-axis, instead, the vertex is either above or below the x-axis, and because the leading coefficient is positive, it opens upward.

Understanding the Quadratic Formula

The quadratic formula is a cornerstone of algebra. It's not just a tool for solving equations; it's a window into the relationships between the coefficients of a quadratic equation and its solutions. By using this formula, we can find out not only the values of x that make the equation true, but also the nature of these solutions. The part under the square root, called the discriminant (b² - 4ac), gives us valuable information about the solutions. If the discriminant is positive, we have two distinct real solutions (as we saw in the first example, X²-3x=0). If it's zero, we have one real solution (a repeated root). If it's negative, as in our second example, we have two complex solutions.

This insight helps us understand the graphical representation of the equation. If the solutions are real, the parabola intersects the x-axis at those points. If the solutions are complex, the parabola doesn't intersect the x-axis, the graph floats either above or below the x-axis. Using the quadratic formula, you gain deeper insights, which is a significant skill for anyone taking on math.

C) Solving the System of Equations: xy + 3 = 5

Alright, let's switch gears and tackle our third equation: xy + 3 = 5. First, simplify the equation. Subtract 3 from both sides: xy = 2. This equation has two variables, x and y, which means we need another equation to find a definite answer for x and y. So, in this instance, we cannot find the exact answer, we can only isolate one variable and express it in terms of the other.

Solving for y, we can divide both sides by x: y = 2/x.

Understanding Variables and Equations

This equation represents a hyperbola. Without a second equation, we can only express one variable in terms of the other. Equations with two variables describe a relationship between those variables, representing an infinite number of possible solutions. Each point (x, y) that satisfies the equation lies on the curve. This is an illustration of how math goes beyond just finding numerical answers, it is a tool for understanding the relationships among variables. Working with equations with multiple variables is important in many areas, such as physics, where we often deal with multiple inter-related quantities.

D) Solving the System of Equations: 3x + 5y = 9

Now, let's explore the last equation: 3x + 5y = 9. Again, we have two variables. To solve, we need another equation to create a system of equations. Since no other equation is given, we can still rearrange the equation to express one variable in terms of the other. We can do it as following:

Subtract 3x from both sides: 5y = 9 - 3x
Divide both sides by 5: y = (9 - 3x) / 5

This equation represents a straight line. Similar to the previous example, without a second equation, we can only express one variable in terms of the other. The line represents the set of all points (x, y) that satisfy the equation. This highlights that an individual equation often has an infinite number of solutions. The key to finding a definitive solution is a system of equations, in which we have as many equations as variables. The point where the lines intersect is the solution to the system.

Understanding Systems of Equations

Systems of equations are fundamental in solving real-world problems. In physics, these systems help solve for motion, force, and energy. In economics, they model supply and demand, and in computer science, they are used to develop algorithms. The process of finding a solution to a system of equations involves finding the values of the variables that satisfy all equations simultaneously. Common methods include substitution, elimination, and graphing. These methods are designed to determine where the lines intersect, providing the solution to the system. The approach taken depends on the equations' form and the required precision. Understanding the concept of systems of equations equips you with the tools to solve complex problems in various fields, from calculating the trajectory of a rocket to optimizing resource allocation in a business environment. The ability to model these problems mathematically and find their solutions is a powerful tool in itself.