Solving Linear Systems How Many Solutions Exist

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Hey guys! Today, we're diving into the fascinating world of linear systems and tackling a question that often pops up in algebra: "How many solutions does this linear system have?" We'll break down the problem step-by-step, making sure you not only get the answer but also understand the underlying concepts. So, grab your pencils, and let's get started!

The Linear System at Hand

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

y = (2/3)x + 2
6x - 4y = -10

Our mission is to figure out whether this system has one solution, no solution, or an infinite number of solutions. Each of these scenarios paints a different picture of how these two lines interact on a graph. Let's explore each possibility in detail.

Understanding Solutions to Linear Systems

Before we jump into solving, let's quickly recap what it means for a linear system to have a solution. A solution to a system of linear equations is a point (x, y) that satisfies both equations simultaneously. Graphically, this point represents the intersection of the two lines. Now, here's where it gets interesting:

  • One Solution: The lines intersect at exactly one point.
  • No Solution: The lines are parallel and never intersect.
  • Infinite Solutions: The lines are the same; they overlap completely.

With this understanding, we can approach our problem with a clearer perspective. Remember, our main keywords here are linear system solutions, so we'll keep that in mind as we proceed.

Solving the Linear System: A Step-by-Step Approach

There are a couple of ways we can tackle this problem, but let's use the substitution method, as it's particularly effective when one equation is already solved for a variable (in our case, the first equation is solved for y). This method helps us find those crucial linear system solutions.

Step 1: Substitute

Since we know that y = (2/3)x + 2, we can substitute this expression for y in the second equation:

6x - 4((2/3)x + 2) = -10

This substitution is a key step in determining the linear system solutions, as it allows us to work with a single variable equation.

Step 2: Simplify and Solve for x

Now, let's simplify the equation and solve for x:

6x - (8/3)x - 8 = -10

To get rid of the fraction, we can multiply the entire equation by 3:

18x - 8x - 24 = -30

Combine like terms:

10x - 24 = -30

Add 24 to both sides:

10x = -6

Divide by 10:

x = -6/10 = -3/5 = -0.6

So, we've found that x = -0.6. This is a significant step towards finding our linear system solutions.

Step 3: Solve for y

Now that we have the value of x, we can plug it back into either of the original equations to find y. Let's use the first equation, as it's simpler:

y = (2/3)(-0.6) + 2

y = (2/3)(-3/5) + 2

y = -2/5 + 2

y = -0.4 + 2

y = 1.6

Therefore, y = 1.6. This gives us a potential solution point for our linear system.

Step 4: Verify the Solution

To be absolutely sure, let's check if the point (-0.6, 1.6) satisfies both original equations. This is crucial for confirming our linear system solutions.

  • Equation 1: y = (2/3)x + 2

    1.  6 = (2/3)(-0.6) + 2
2.  6 = -0.4 + 2
3.  6 = 1.6  (This holds true)

  • Equation 2: 6x - 4y = -10

    6(-0.6) - 4(1.6) = -10
-3.6 - 6.4 = -10
-10 = -10  (This also holds true)

Since the point (-0.6, 1.6) satisfies both equations, it is indeed a solution to the linear system. This is a confirmed linear system solution.

Determining the Number of Solutions

We've found a solution! But does this mean we're done? Not quite. We still need to determine how many solutions the system has. Since we've found one solution, we know it's not "no solution" (parallel lines). To determine if there are infinite solutions, we would need to check if the two equations represent the same line. However, since we found a unique solution (-0.6, 1.6), we can confidently say that the system has one solution. Understanding how to find linear system solutions is key here.

Analyzing the Options

Now, let's look at the options provided:

A. one solution: (-0.6,-1.6) B. one solution: (-0.6,1.6) C. no solution D. infinite number of solutions

Based on our calculations, the correct answer is B. one solution: (-0.6, 1.6). We found that the lines intersect at the point (-0.6, 1.6), confirming that there is exactly one solution. This entire process has been about finding linear system solutions.

Alternative Approach: Slope-Intercept Form

Another way to determine the number of solutions is to rewrite both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. This method provides a visual understanding of linear system solutions.

Equation 1: Already in Slope-Intercept Form

The first equation is already in slope-intercept form:

y = (2/3)x + 2

Here, the slope m is 2/3, and the y-intercept b is 2.

Equation 2: Converting to Slope-Intercept Form

Let's rewrite the second equation:

6x - 4y = -10

Subtract 6x from both sides:

-4y = -6x - 10

Divide by -4:

y = (3/2)x + 5/2

Here, the slope m is 3/2, and the y-intercept b is 5/2. Analyzing slopes and intercepts is a powerful way to understand linear system solutions.

Comparing Slopes and Intercepts

Now, let's compare the slopes and y-intercepts:

  • Equation 1: Slope = 2/3, y-intercept = 2
  • Equation 2: Slope = 3/2, y-intercept = 5/2

Since the slopes are different (2/3 ≠ 3/2), the lines are not parallel and will intersect at exactly one point. This confirms that there is one solution. This comparison highlights the importance of understanding slopes when determining linear system solutions.

Key Takeaways and Strategies for Solving Linear Systems

So, we've successfully navigated through this problem and found that the linear system has one solution. But what are the key takeaways and strategies we can use for similar problems? Let's recap:

1. Understand the Solution Types

  • One Solution: Lines intersect at one point.
  • No Solution: Lines are parallel (same slope, different y-intercepts).
  • Infinite Solutions: Lines are the same (same slope, same y-intercept).

Recognizing these scenarios is the first step in finding linear system solutions.

2. Choose the Right Method

  • Substitution: Useful when one equation is solved for a variable.
  • Elimination: Useful when coefficients of one variable are opposites or easily made opposites.
  • Graphing: Useful for visualizing the system and estimating solutions.

Selecting the appropriate method can significantly simplify the process of finding linear system solutions.

3. Verify Your Solution

Always plug your solution back into both original equations to ensure it satisfies the system. This step is crucial for avoiding errors and confirming your linear system solutions.

4. Slope-Intercept Form

Rewriting equations in slope-intercept form (y = mx + b) allows for quick comparison of slopes and y-intercepts, helping determine the number of solutions. This method is particularly helpful in visualizing linear system solutions.

5. Practice Makes Perfect

The more you practice solving linear systems, the more comfortable and confident you'll become. Try different types of problems and methods to hone your skills in finding linear system solutions.

Conclusion: Mastering Linear System Solutions

Alright, guys! We've covered a lot of ground today, from understanding the basics of linear systems to solving a specific problem and discussing strategies for future challenges. Remember, the key to mastering linear system solutions is understanding the concepts, choosing the right methods, and practicing consistently.

So, next time you encounter a linear system, you'll be well-equipped to tackle it with confidence. Keep practicing, and you'll become a pro at finding those elusive solutions! Now go out there and conquer those equations!