Solving Linear Systems How To Determine The Number Of Solutions

Navigating the world of linear systems often involves determining the number of solutions a given system possesses. This article dives deep into how to determine the number of solutions for a given linear system, using the example provided:

 y = \frac{2}{3}x + 2 \
 6x - 4y = -10

We will explore various methods to analyze this system, including substitution, elimination, and graphical approaches, to arrive at the correct answer. We'll also discuss what each type of solution—one solution, no solution, or infinite solutions—means geometrically.

Understanding Linear Systems and Their Solutions

To properly address the question, we must first grasp the fundamentals of linear systems. A linear system is a set of two or more linear equations involving the same variables. The solution to a linear system is the set of values for the variables that satisfy all equations simultaneously. Graphically, each linear equation represents a line, and the solution to the system corresponds to the point(s) where these lines intersect.

There are three possibilities for the number of solutions in a linear system:

1. One Solution: The lines intersect at exactly one point. This means there is a unique pair of values (x, y) that satisfies both equations.
2. No Solution: The lines are parallel and never intersect. This indicates that there is no pair of values (x, y) that can satisfy both equations simultaneously.
3. Infinite Solutions: The lines are coincident, meaning they are the same line. Every point on the line represents a solution to the system, resulting in an infinite number of solutions.

Understanding these possibilities is crucial for solving linear systems effectively. We will apply these concepts to the given system to find its solution.

Analyzing the Given Linear System

Let's revisit the system of equations:

 y = \frac{2}{3}x + 2 \
 6x - 4y = -10

To determine the number of solutions, we can use several methods, including substitution, elimination, and graphical methods. We'll start by using the substitution method because the first equation is already solved for y. This will allow us to easily substitute the expression for y into the second equation.

Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. In our case, the first equation is already solved for y:

 y = \frac{2}{3}x + 2

Now, we substitute this expression for y into the second equation:

 6x - 4(\frac{2}{3}x + 2) = -10

Next, we simplify and solve for x:

 6x - \frac{8}{3}x - 8 = -10 \
 (6 - \frac{8}{3})x = -10 + 8 \
 (\frac{18}{3} - \frac{8}{3})x = -2 \
 \frac{10}{3}x = -2

Multiply both sides by ${\frac{3}{10}}$ to isolate x:

 x = -2 \cdot \frac{3}{10} \
 x = -\frac{6}{10} \
 x = -\frac{3}{5} \
 x = -0.6

Now that we have the value of x, we can substitute it back into the first equation to find the value of y:

 y = \frac{2}{3}(-0.6) + 2 \
 y = \frac{2}{3}(-\frac{3}{5}) + 2 \
 y = -\frac{2}{5} + 2 \
 y = -0.4 + 2 \
 y = 1.6

So, the solution to the system is x = -0.6 and y = 1.6. This means the lines intersect at the point (-0.6, 1.6), indicating that there is exactly one solution.

Elimination Method

Another approach to solving linear systems is the elimination method. This method involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. To use this method, we first need to rewrite the first equation in the standard form (Ax + By = C):

 y = \frac{2}{3}x + 2

Multiply both sides by 3 to eliminate the fraction:

 3y = 2x + 6

Rearrange the equation to get the standard form:

 -2x + 3y = 6

Now, we have the following system:

 -2x + 3y = 6 \
 6x - 4y = -10

To eliminate x, we can multiply the first equation by 3:

 3(-2x + 3y) = 3(6) \
 -6x + 9y = 18

Now, add the modified first equation to the second equation:

 (-6x + 9y) + (6x - 4y) = 18 + (-10) \
 5y = 8

Divide by 5 to solve for y:

 y = \frac{8}{5} \
 y = 1.6

Substitute y = 1.6 back into one of the original equations, such as -2x + 3y = 6:

 -2x + 3(1.6) = 6 \
 -2x + 4.8 = 6 \
 -2x = 1.2 \
 x = -0.6

Again, we find the solution to be x = -0.6 and y = 1.6, confirming that there is one solution.

Graphical Method

The graphical method involves plotting the two lines on a coordinate plane and finding their point of intersection. The point of intersection represents the solution to the system. Let’s rewrite the equations to make them easier to graph:

1. ${ y = \frac{2}{3}x + 2 }$
2. ${ 6x - 4y = -10 }$

For the second equation, we can solve for y:

 -4y = -6x - 10 \
 y = \frac{3}{2}x + \frac{5}{2}

Now we have two equations in slope-intercept form:

1. ${ y = \frac{2}{3}x + 2 }$
2. ${ y = \frac{3}{2}x + \frac{5}{2} }$

By graphing these lines, we would see that they intersect at one point. The first line has a slope of ${\frac{2}{3}}$ and a y-intercept of 2. The second line has a slope of ${\frac{3}{2}}$ and a y-intercept of ${\frac{5}{2}}$ (or 2.5).

Since the slopes are different, the lines are not parallel and will intersect at exactly one point. This confirms that the system has one solution. The graphical representation further reinforces the algebraic solutions we found using the substitution and elimination methods.

Identifying the Correct Answer

Based on our analysis using the substitution, elimination, and graphical methods, we have consistently found that the system has one solution: x = -0.6 and y = 1.6. This corresponds to option B in the given choices.

Therefore, the correct answer is:

• B. One solution: (-0.6, 1.6)

This solution satisfies both equations in the system, making it the unique point of intersection for the two lines.

Implications of Different Solution Types

Understanding the implications of each type of solution is important for a comprehensive grasp of linear systems. Let’s briefly discuss each case:

• One Solution: As demonstrated in our example, this occurs when the lines have different slopes and intersect at a single point. This is the most common scenario in linear systems.

• No Solution: This happens when the lines are parallel, meaning they have the same slope but different y-intercepts. Parallel lines never intersect, so there is no solution that satisfies both equations. An example of such a system would be:

   y = 2x + 3 \
   y = 2x - 1
  

  These lines have the same slope (2) but different y-intercepts (3 and -1), so they are parallel and have no intersection.

• Infinite Solutions: This occurs when the two equations represent the same line. In other words, one equation is a multiple of the other. In this case, every point on the line is a solution to the system. An example is:

   y = x + 1 \
   2y = 2x + 2
  

  The second equation is simply twice the first equation, so they represent the same line. Any point that satisfies the first equation will also satisfy the second, leading to infinite solutions.

Conclusion

In summary, determining the number of solutions for a linear system involves analyzing the equations to see if they intersect at one point, are parallel (no intersection), or are the same line (infinite intersections). By using methods like substitution, elimination, and graphical analysis, we can accurately determine the number of solutions.

For the given system:

 y = \frac{2}{3}x + 2 \
 6x - 4y = -10

We found that there is one solution at the point (-0.6, 1.6). This was achieved by using the substitution method and verifying with the elimination method. The graphical method further supported this conclusion by showing that the two lines intersect at a single point.

Understanding these concepts is crucial for solving more complex problems in linear algebra and related fields. Whether you are dealing with linear equations in mathematics, physics, or engineering, the ability to determine the number of solutions is a fundamental skill.

By mastering these techniques, you can confidently tackle any linear system and accurately determine its solution set. The key is to practice and understand the underlying principles that govern these systems.