Solving Linear Systems: How Many Solutions?

Hey guys! Let's dive into figuring out how many solutions a system of linear equations can have. We're given two equations:

1. $y = -\frac{1}{2}x + 4$
2. $x + 2y = -8$

Our mission, should we choose to accept it, is to determine whether this system has one solution, no solution, or an infinite number of solutions. Let's get started!

Understanding Linear Systems

Before we jump into solving, let's quickly recap what a system of linear equations is all about. Basically, it's a set of two or more linear equations that we're solving together. The solution to such a system is the set of values that satisfy all the equations in the system simultaneously. Graphically, each linear equation represents a straight line, and the solution to the system is the point where these lines intersect. If the lines never meet, there is no solution. If the lines are the same, there are infinite solutions. This visual intuition is super helpful!

When we talk about the number of solutions, we have three possible scenarios:

• One Solution: The lines intersect at exactly one point. This means there's a unique pair of (x, y) values that satisfy both equations.
• No Solution: The lines are parallel and never intersect. This means there's no pair of (x, y) values that can satisfy both equations simultaneously.
• Infinite Solutions: The lines are identical, meaning they overlap completely. Every point on one line is also on the other line, so there are infinitely many (x, y) pairs that satisfy both equations.

To figure out which scenario our system falls into, we'll use a bit of algebra to manipulate the equations and see what happens.

Solving the System

Okay, let's roll up our sleeves and solve this system. The first equation, $y = -\frac{1}{2}x + 4$, is already nicely solved for y. This is great because we can use the substitution method. We'll substitute the expression for y from the first equation into the second equation.

The second equation is $x + 2y = -8$. Now, replace y with $(-\frac{1}{2}x + 4)$:

$x + 2(-\frac{1}{2}x + 4) = -8$

Now, let's simplify this equation. Distribute the 2:

$x - x + 8 = -8$

Notice something interesting? The x terms cancel each other out! We're left with:

$8 = -8$

Whoa! This is definitely not true. 8 does not equal -8. This is a contradiction. What does this mean for our system?

Interpreting the Result

The fact that we arrived at a contradiction ($8 = -8$) tells us that the system has no solution. Remember, we were trying to find values for x and y that would satisfy both equations simultaneously. But the algebra led us to a statement that is always false, regardless of the values of x and y. This means that the two lines represented by the equations never intersect.

Geometrically, these lines are parallel. They have the same slope but different y-intercepts. Let's quickly confirm this by rewriting the second equation in slope-intercept form (y = mx + b).

Starting with $x + 2y = -8$, subtract x from both sides:

$2y = -x - 8$

Now, divide both sides by 2:

$y = -\frac{1}{2}x - 4$

Notice that this equation, $y = -\frac{1}{2}x - 4$, has the same slope ($-\frac{1}{2}$) as the first equation, $y = -\frac{1}{2}x + 4$, but a different y-intercept (4 vs. -4). This confirms that the lines are parallel and therefore do not intersect.

The Answer

So, after all that algebraic maneuvering, we've determined that the system of equations has no solution. The correct answer is:

C. no solution

Key Takeaways

• A system of linear equations can have one solution, no solution, or an infinite number of solutions.
• If solving a system leads to a contradiction (like $8 = -8$), the system has no solution.
• Parallel lines (lines with the same slope but different y-intercepts) represent a system with no solution.
• Identical lines represent a system with infinitely many solutions.
• The substitution method involves solving one equation for one variable and substituting that expression into the other equation.

Extra Practice

Want to sharpen your skills? Here are some additional exercises you can try:

1. Solve the system:

   $y = 2x - 1$

   $4x - 2y = 2$

   How many solutions does it have?

2. Solve the system:

   $x + y = 5$

   $x - y = 1$

   What is the solution?

3. Determine whether the following system has one solution, no solution, or infinitely many solutions, without actually solving it:

   $3x + 6y = 9$

   $x + 2y = 3$

These exercises will give you more practice in recognizing the different scenarios and applying the appropriate methods to solve linear systems. Good luck, and keep on solving!

Conclusion

Alright, folks! That wraps up our exploration of determining the number of solutions in a linear system. Remember, the key is to manipulate the equations, watch for contradictions or identities, and understand the geometric interpretation of the lines. Keep practicing, and you'll become a pro at solving linear systems in no time!