No Solution System? Math Equations Explained!

by Kenji Nakamura 46 views

Hey guys! Let's dive into a super interesting math problem today. We're going to figure out how many solutions there are for a system of equations. This is a classic algebra question, and once you get the hang of it, you'll be solving these like a pro. So, grab your thinking caps, and let's get started!

The System of Equations

First, let's take a look at the system of equations we're working with:

\left\{
\begin{array}{l}
3 x+y=18 \\
3 x+y=16
\end{array}
\right.

We have two equations here, both involving x and y. Our goal is to find out if there are any values for x and y that satisfy both equations simultaneously. This is where the fun begins!

Understanding Systems of Equations

Before we jump into solving, let's quickly recap what a system of equations actually represents. Think of each equation as a line on a graph. The solutions to the equation are the points that lie on the line. When we have a system of two equations, we're essentially looking for the points where the two lines intersect. These intersection points are the solutions that satisfy both equations.

There are a few possibilities:

  • One solution: The lines intersect at one point.
  • No solution: The lines are parallel and never intersect.
  • Infinitely many solutions: The lines are the same (they overlap).

Now that we've got the basics down, let's analyze our specific system.

Analyzing the Equations

Okay, let's really dig into these equations. We've got:

  • Equation 1: 3x + y = 18
  • Equation 2: 3x + y = 16

Notice anything interesting? Look closely! The left-hand side of both equations is identical (3x + y), but the right-hand sides are different (18 and 16). This is a huge clue.

The Key Observation

This is where the magic happens. Think about it: if 3x + y equals 18 in the first equation, can it also equal 16 in the second equation? The answer is a resounding no! There's no way that the same expression (3x + y) can simultaneously equal two different values.

Visualizing the Problem

Let's imagine these equations as lines on a graph. If we were to rewrite these equations in slope-intercept form (y = mx + b), we'd see something like this:

  • Equation 1: y = -3x + 18
  • Equation 2: y = -3x + 16

Notice that both lines have the same slope (-3), but different y-intercepts (18 and 16). What does this mean geometrically? It means the lines are parallel!

Parallel lines, as you might remember from geometry, never intersect. And since the solutions to the system are the intersection points, this means there are no solutions for this system of equations. Cool, right?

Determining the Number of Solutions

Now, let's solidify our understanding. When we look at a system of equations, there are a few ways to determine the number of solutions without even solving them completely:

1. Comparing Coefficients

This is the method we used in our problem. If the coefficients of x and y are proportional but the constant terms are not, the lines are parallel, and there are no solutions. Let's break this down:

  • Proportional Coefficients: In our case, the ratio of the coefficients of x (3 and 3) is the same as the ratio of the coefficients of y (1 and 1). They're both 1:1.
  • Non-Proportional Constants: The ratio of the constants (18 and 16) is not the same as the ratio of the coefficients. This is our red flag!

2. Solving the System (and What Happens)

We could try to solve the system using methods like substitution or elimination. Let's see what happens if we try elimination:

Subtract Equation 2 from Equation 1:

(3x + y) - (3x + y) = 18 - 16
0 = 2

Wait a minute... 0 = 2? That's definitely not true! This contradiction confirms that there are no solutions to the system. When you try to solve a system and end up with a false statement, that's a clear sign that the lines are parallel.

3. Graphical Interpretation

As we discussed earlier, visualizing the equations as lines on a graph can be incredibly helpful. If you graph the two equations, you'll see two parallel lines. Parallel lines never intersect, meaning there are no common points (solutions).

The Answer: A. None

So, after analyzing the equations, understanding the concept of parallel lines, and even attempting to solve the system, we've arrived at the answer. There are no solutions for the system of equations:

\left\{
\begin{array}{l}
3 x+y=18 \\
3 x+y=16
\end{array}
\right.

The correct answer is A. none.

Why Other Options Are Incorrect

Let's quickly discuss why the other options aren't correct:

  • B. one: This would be true if the lines intersected at a single point.
  • C. two: Lines can't intersect at exactly two points. They either intersect at one point, infinitely many points (if they're the same line), or no points (if they're parallel).
  • D. infinitely many: This would be true if the two equations represented the same line.

Real-World Applications

You might be thinking,