Solving Systems Of Equations: A Detailed Explanation

by Admin 53 views
Solving Systems of Equations: A Detailed Explanation

Let's dive into the world of systems of equations! We're going to break down a problem step-by-step, making sure everything is crystal clear. Our main goal here is to figure out which statement is true about the given system of equations. So, buckle up, and let's get started!

Understanding the Equations

The system of equations we're dealing with looks like this:

y = -2x + 4
3y + x = -3

Before we jump into analyzing these equations, let's quickly recap what systems of equations are and why they're so important. A system of equations is just a set of two or more equations that share variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Think of it like finding the sweet spot that works for every equation in the group.

Systems of equations pop up all over the place in real-world applications. For example, they can help you figure out the break-even point for a business, plan efficient routes for delivery services, or even model complex relationships in scientific research. Mastering systems of equations opens up a whole new world of problem-solving possibilities!

Equation 1: y = -2x + 4

The first equation, y = -2x + 4, is already in a friendly format. This form is known as slope-intercept form, which is super handy because it tells us two key things right away: the slope and the y-intercept. The general slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

In our case, m = -2, which means the slope of the line is -2. This tells us that for every 1 unit we move to the right along the x-axis, the line goes down by 2 units along the y-axis. The b = 4, meaning the y-intercept is 4. This is the point where the line crosses the y-axis, so it's the point (0, 4).

Equation 2: 3y + x = -3

The second equation, 3y + x = -3, is a bit more mysterious. It's not in slope-intercept form, so we can't immediately read off the slope and y-intercept. To get it into slope-intercept form, we need to isolate y on one side of the equation. Here’s how we can do it:

  1. Subtract x from both sides: 3y = -x - 3
  2. Divide both sides by 3: y = (-1/3)x - 1

Now, the equation is in slope-intercept form: y = (-1/3)x - 1. We can now see that the slope is -1/3 and the y-intercept is -1. This line has a gentler downward slope than the first one, and it crosses the y-axis at the point (0, -1).

Analyzing the Given Statements

Now that we understand both equations, let's look at the statements provided and see which one holds true.

Statement A: Both Equations Are in Slope-Intercept Form

This statement is partially true. The first equation, y = -2x + 4, is indeed in slope-intercept form. However, the second equation, 3y + x = -3, is not initially in slope-intercept form. We had to convert it. So, statement A is not entirely correct.

Statement B: The First Equation Converted to Slope-Intercept Form is y + 2x = 4

Let's analyze this statement. The first equation is already given as y = -2x + 4. If we rearrange this equation to match the form in the statement, we would add 2x to both sides:

y + 2x = -2x + 4 + 2x
y + 2x = 4

So, the first equation converted to that form is indeed y + 2x = 4. Therefore, statement B is true.

Why Statement B is the Correct Answer

Statement B accurately describes a valid transformation of the first equation into an equivalent form. It reflects a simple algebraic manipulation that preserves the equation's integrity.

Additional Insights on Solving Systems of Equations

When solving systems of equations, there are a few common methods you can use:

  • Substitution: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination: Add or subtract the equations to eliminate one variable.
  • Graphing: Plot both equations on a graph and find the point where they intersect. This point represents the solution to the system.

Each method has its strengths and weaknesses, and the best choice depends on the specific equations you're dealing with. For instance, substitution works well when one equation is already solved for a variable, while elimination is great when the coefficients of one variable are opposites.

Visualizing the Solution

To truly understand what's going on with these equations, it's helpful to visualize them. When you graph the two equations, you'll see two lines. The point where these lines intersect is the solution to the system of equations. At that point, both equations are satisfied simultaneously.

In our example, the lines y = -2x + 4 and y = (-1/3)x - 1 intersect at a specific point. Finding this point gives us the values of x and y that make both equations true. You can find this point by setting the two equations equal to each other and solving for x:

-2x + 4 = (-1/3)x - 1

Solving this equation will give you the x-coordinate of the intersection point. Then, you can plug that value back into either equation to find the y-coordinate.

Conclusion

In summary, after analyzing the given system of equations and the provided statements, we can confidently conclude that statement B is the correct one. The first equation, y = -2x + 4, can indeed be converted to the form y + 2x = 4 through a simple algebraic manipulation. Understanding the different forms of linear equations and how to manipulate them is crucial for solving systems of equations and tackling real-world problems.

Remember, practice makes perfect! The more you work with systems of equations, the more comfortable and confident you'll become. Keep exploring, keep learning, and keep solving!