System Of Equations: How Many Solutions?

Hey guys! Today, we're diving deep into the super interesting world of systems of equations. Specifically, we're tackling a question that pops up a lot: how many solutions does a given system have? It might sound a bit daunting, but trust me, once you get the hang of it, it's pretty straightforward. We're going to use the system you've presented,

$\begin{array}{l} 7 x-y=-1 \\ -5 x-2 y=11 \end{array}$

and break down exactly how to figure out the number of solutions. This is a fundamental concept in algebra, and understanding it will open doors to solving all sorts of problems, from simple line intersections on a graph to more complex real-world scenarios.

So, what exactly does it mean for a system of equations to have solutions? Think of each equation as a line on a graph. A solution to the system is any point (x, y) that lies on both lines. If the lines intersect at a single point, there's exactly one solution. If the lines are parallel and never meet, there are no solutions. And if the lines are actually the same line, then every single point on that line is a solution, meaning there are infinitely many solutions. Our main goal today is to determine which of these three cases applies to our specific system.

We've got two main methods to find out how many solutions exist: the substitution method and the elimination method. Both are fantastic tools, and sometimes one is easier than the other depending on the equations. We can also visualize this by graphing the lines, but honestly, for finding the number of solutions without needing the exact solution itself, algebraic methods are usually quicker and more precise. We'll walk through these methods step-by-step, so don't worry if you're new to this. By the end, you'll be a pro at spotting the number of solutions for any system of two linear equations!

Understanding the Basics: What is a System of Equations?

Alright, let's set the stage properly. Before we jump into solving, it's crucial to get a solid grip on what a system of equations actually is. Think of it as a team of equations, usually two or more, that share the same variables. In our case, we're dealing with a system of two linear equations with two variables, x and y. Each equation represents a line in a two-dimensional coordinate plane. The magic happens when we look for values of x and y that satisfy all the equations in the system simultaneously. This is what we call a solution to the system.

Imagine you're plotting these equations on graph paper. The first equation, 7x - y = -1, will draw a straight line. The second equation, -5x - 2y = 11, will draw another straight line. The solutions to the system are the points where these two lines intersect. Now, how many times can two straight lines intersect? This is the core question that helps us determine the number of solutions.

1. One Solution: This happens when the two lines are distinct and not parallel. They will cross each other at exactly one point. This point is your unique solution (x, y) that makes both equations true.
2. No Solutions: This occurs when the two lines are parallel but different. Parallel lines have the same slope but different y-intercepts. Since they never meet, there's no point that lies on both lines, hence no solution.
3. Infinitely Many Solutions: This is the case when the two equations actually represent the same line. They might look different initially, but if you manipulate one equation, you'll find it's just a scaled version of the other. In this scenario, every single point on the line is a solution because it satisfies both equations.

So, when we're asked "how many solutions does the system have?", we're really asking: Do these two lines intersect once, never, or are they the same line? Our job is to use algebraic techniques to figure this out without necessarily finding the actual point(s) of intersection, unless that's part of the process.

Understanding these three possibilities is key. It's like knowing the different outcomes before a coin flip: heads, tails, or maybe it lands on its edge (though that's super rare!). With linear systems, these three outcomes are the only ones possible. We'll explore how to test for each one using our example system, making sure you feel confident and ready to tackle similar problems.

Method 1: Substitution - The Sneaky Solver

Let's get hands-on with our first powerful technique: the substitution method. This is a fantastic way to solve systems of equations, and it's especially useful for determining the number of solutions. The basic idea is to isolate one variable in one equation and then substitute that expression into the other equation. This process reduces a two-variable system into a single equation with just one variable, which is much easier to handle.

Our system is:

$\begin{array}{l} 7 x-y=-1 \quad (1) \\ -5 x-2 y=11 \quad (2) \end{array}$

Looking at equation (1), 7x - y = -1, it seems pretty easy to isolate y. We can add y to both sides and then add 1 to both sides. Let's do that:

7x + 1 = y

So, we now know that y is equivalent to 7x + 1. This is the expression we're going to substitute into equation (2). Wherever we see y in equation (2), we'll replace it with (7x + 1). Remember to use parentheses to avoid sign errors!

Equation (2) is: -5x - 2y = 11

Substitute y = 7x + 1 into equation (2):

-5x - 2(7x + 1) = 11

Now, let's simplify and solve for x:

-5x - 14x - 2 = 11

Combine the x terms:

-19x - 2 = 11

Add 2 to both sides:

-19x = 13

Divide by -19:

x = -13/19

Aha! We found a specific value for x. This is a huge clue! When the substitution method leads to a unique numerical value for a variable, it means there is exactly one solution to the system. Why? Because we were able to isolate a variable and substitute it, leading to a solvable equation for the other variable. If we were to continue and substitute this x value back into y = 7x + 1, we'd find a unique y value as well, giving us our single intersection point.

Now, what if things had gone differently? Let's imagine a scenario where, after substitution and simplification, we ended up with something like 0 = 5. This statement is false. It means that no matter what values x and y take, this equation can never be true. In such a case, it implies that the original lines never intersect, and the system has no solutions. Conversely, if we had ended up with 0 = 0 (or any other true statement like 5 = 5), this means the equation is true for all values of x (and consequently y). This happens when the two original equations represent the same line, leading to infinitely many solutions.

Since we got a specific value for x (x = -13/19), we know for sure that our system has exactly one solution. The substitution method is a reliable way to get to this answer, guys. It systematically simplifies the problem until you can see the nature of the solution.

Method 2: Elimination - The Balanced Approach

Next up, let's explore the elimination method, also known as the addition method. This technique is another powerful tool in our algebra arsenal for solving systems of equations and determining the number of solutions. The core idea here is to manipulate one or both equations (by multiplying them by constants) so that when you add or subtract the equations, one of the variables cancels out (is eliminated).

Our system is:

$\begin{array}{l} 7 x-y=-1 \quad (1) \\ -5 x-2 y=11 \quad (2) \end{array}$

Our goal is to make the coefficients of either x or y opposites. Let's look at the y coefficients: -1 in equation (1) and -2 in equation (2). If we multiply equation (1) by -2, the y coefficient will become +2, which is the opposite of -2 in equation (2).

Multiply equation (1) by -2:

-2 * (7x - y) = -2 * (-1)

This gives us:

-14x + 2y = 2 \quad (1')

Now, we have our modified system:

$\begin{array}{l} -14 x+2 y=2 \quad (1') \\ -5 x-2 y=11 \quad (2) \end{array}$

Notice that the y coefficients are now +2 and -2. If we add equation (1') and equation (2) together, the y terms will cancel out:

(-14x + 2y) + (-5x - 2y) = 2 + 11

Combine like terms:

-14x - 5x + 2y - 2y = 13

-19x + 0y = 13

-19x = 13

Divide by -19:

x = -13/19

Bingo! Just like with the substitution method, we've arrived at a specific, unique value for x. This outcome, where we successfully eliminate one variable and end up with a solvable equation for the other, is a definitive indicator that the system has exactly one solution. The two lines represented by these equations intersect at a single point.

Think about the other possibilities with elimination. If, after attempting to eliminate a variable, you ended up with a statement like 0 = 5 (a false statement), it would mean that the original equations are inconsistent and represent parallel lines that never meet. Therefore, the system would have no solutions. On the flip side, if you ended up with a statement like 0 = 0 (a true statement), it signifies that the two equations are dependent and represent the same line, leading to infinitely many solutions.

Because our elimination process yielded a concrete value for x, we can confidently conclude that our system:

$\begin{array}{l} 7 x-y=-1 \\ -5 x-2 y=11 \end{array}$

has exactly one solution. Both substitution and elimination methods have led us to the same conclusion, reinforcing its accuracy. This balanced approach of elimination is super handy, especially when the equations aren't set up perfectly for substitution.

Visualizing the Solutions: The Graphing Method

While algebraic methods like substitution and elimination are usually the most efficient for determining the number of solutions, it's incredibly helpful to understand how this relates to the graphing method. Visualizing the lines on a coordinate plane really solidifies the concept of solutions as intersection points.

Let's take our system:

$\begin{array}{l} 7 x-y=-1 \quad (1) \\ -5 x-2 y=11 \quad (2) \end{array}$

To graph these, we typically want them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

For equation (1): 7x - y = -1

Add y to both sides and add 1 to both sides:

y = 7x + 1

This line has a slope of m = 7 and a y-intercept of b = 1. So, it crosses the y-axis at (0, 1) and rises steeply.

For equation (2): -5x - 2y = 11

Subtract -5x from both sides (which means adding 5x):

-2y = 5x + 11

Divide everything by -2:

y = (5x / -2) + (11 / -2)

y = -5/2 * x - 11/2

This line has a slope of m = -5/2 (or -2.5) and a y-intercept of b = -11/2 (or -5.5). This line has a negative slope, meaning it goes downwards as you move to the right, and it crosses the y-axis lower down.

Now, let's think about the slopes:

• Line 1 has a slope of 7.
• Line 2 has a slope of -5/2.

Since the slopes are different (7 ≠ -5/2), these two lines are not parallel. Non-parallel lines in a plane are guaranteed to intersect at exactly one point. This single intersection point is the coordinates (x, y) that satisfies both equations simultaneously.

This graphical perspective directly confirms our findings from the substitution and elimination methods. The different slopes tell us visually that there will be a unique point where these lines cross.

What if the slopes were the same?

• If the slopes were the same AND the y-intercepts were the same, the equations would represent the exact same line. Every point on that line would be a solution, giving us infinitely many solutions.
• If the slopes were the same BUT the y-intercepts were different, the equations would represent parallel lines. These lines would never intersect, meaning there would be no solutions.

So, by comparing the slopes (and potentially y-intercepts if the slopes are equal), the graphing method provides a clear, visual confirmation of the number of solutions. It’s a great way to build intuition for why a system has one, none, or infinite solutions. For our specific system, the different slopes (7 and -5/2) clearly indicate that there is exactly one solution.

Conclusion: How Many Solutions Does Our System Have?

Alright, guys, we've journeyed through the different ways to determine the number of solutions for a system of linear equations. We've applied the substitution method, the elimination method, and even considered the visual aspect of graphing. Each method has brought us to the same definitive conclusion for our particular system:

$\begin{array}{l} 7 x-y=-1 \\ -5 x-2 y=11 \end{array}$

Through substitution, we isolated y from the first equation (y = 7x + 1) and plugged it into the second. This led us to -19x = 13, which gave us a unique value for x. This outcome signals exactly one solution.

Using elimination, we multiplied the first equation by -2 to make the y coefficients opposites. Adding the modified equations resulted in -19x = 13 again, yielding a specific x value. This also confirms exactly one solution.

Graphically, we found that the first equation simplifies to y = 7x + 1 (slope = 7, y-intercept = 1), and the second simplifies to y = -5/2 * x - 11/2 (slope = -5/2, y-intercept = -11/2). Since the slopes are different (7 ≠ -5/2), the lines are not parallel and must intersect at a single point. This visual confirmation also points to exactly one solution.

Therefore, the answer to the question, "How many solutions does the system of equations below have?" is unequivocally one solution. It's fantastic when all methods align, right? It really builds confidence in your answer. Remember these techniques, and you'll be able to confidently tackle any system of linear equations that comes your way, whether it's in a math class, a test, or even a real-world problem!