Solving A System Of Equations: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of mathematics to tackle a system of equations. Specifically, we'll be solving the following system:

$$\left\{\begin{array}{l} y=\frac{1}{2} x+\frac{3}{4} \\ -6 x+12 y=-47 \end{array}\right.$$

Systems of equations pop up everywhere, from calculating the best deals at the grocery store to designing complex engineering structures. Understanding how to solve them is a crucial skill, so let's break it down step by step!

Understanding Systems of Equations

Before we jump into solving, let's make sure we're all on the same page about what a system of equations actually is. Simply put, it's a set of two or more equations that share the same variables. The goal is to find values for those variables that satisfy all the equations in the system simultaneously. Think of it like finding a secret code that unlocks all the equations at once. There are generally three possible outcomes when solving a system:

• One unique solution: This means there's only one set of values for the variables that makes all the equations true.
• No solution: This happens when the equations contradict each other. There are no values for the variables that can satisfy all equations at the same time. The lines represented by the equations are parallel and never intersect.
• Infinitely many solutions: This occurs when the equations are essentially the same, just written differently. Any solution to one equation is also a solution to the other. The lines represented by the equations are coincident (they are the same line).

In our case, we have two equations with two variables, x and y. Our mission, should we choose to accept it (and we do!), is to find the values of x and y that make both equations true. Let’s get started!

Choosing a Solution Method

There are a few common methods for solving systems of equations, including:

• Substitution: Solve one equation for one variable and substitute that expression into the other equation.
• Elimination (or Addition): Manipulate the equations so that when you add or subtract them, one of the variables cancels out.
• Graphing: Graph both equations and find the point where they intersect. This method is useful for visualizing the solution, but it's not always the most accurate, especially if the solutions aren't integers.

For this particular system, substitution looks like the easiest route because the first equation is already solved for y. This means we can directly substitute the expression for y from the first equation into the second equation. This will leave us with one equation in terms of x only, which we can then easily solve.

Solving by Substitution: Step-by-Step

Okay, let's get our hands dirty! Here's how we'll solve the system using substitution:

1. Identify the Equations:

   We have:

   • Equation 1: $y = \frac{1}{2}x + \frac{3}{4}$
   • Equation 2: $-6x + 12y = -47$

2. Substitute:

   Since Equation 1 is already solved for y, we'll substitute the expression $\frac{1}{2}x + \frac{3}{4}$ for y in Equation 2:

   $-6x + 12(\frac{1}{2}x + \frac{3}{4}) = -47$

3. Simplify and Solve for x:

   Now we have an equation with only one variable, x. Let's simplify and solve it:

   • $-6x + 6x + 9 = -47$
   • $9 = -47$ Wait a minute! This doesn't look right.

4. Analyze the Result

   We arrived at the statement $9 = -47$, which is clearly false. This means that there is no solution to this system of equations. The two equations represent lines that are parallel and never intersect. Therefore, no values of x and y can satisfy both equations simultaneously.

Why No Solution?

It's always good to understand why a system might have no solution. In this case, let's rewrite the second equation in slope-intercept form (y = mx + b) to compare it with the first equation.

Starting with $-6x + 12y = -47$, we can solve for y:

1. Add 6x to both sides: $12y = 6x - 47$
2. Divide both sides by 12: $y = \frac{1}{2}x - \frac{47}{12}$

Now we can clearly see the two equations in slope-intercept form:

• Equation 1: $y = \frac{1}{2}x + \frac{3}{4}$
• Equation 2 (rewritten): $y = \frac{1}{2}x - \frac{47}{12}$

Notice that both equations have the same slope ($\frac{1}{2}$), but different y-intercepts ($\frac{3}{4}$ and $-\frac{47}{12}$). This means the lines are parallel and will never intersect, confirming that there is no solution to the system.

Key Takeaways

• Systems of equations can have one solution, no solutions, or infinitely many solutions.
• Substitution is a powerful method for solving systems when one equation is already solved for a variable.
• If you arrive at a contradiction (like 9 = -47) when solving, the system has no solution.
• Parallel lines (lines with the same slope but different y-intercepts) represent systems with no solutions.

So, there you have it! Even though this system had no solution, we learned valuable techniques for solving systems of equations in general. Keep practicing, and you'll become a system-solving superstar in no time!