Identify the solution. Plug the solution into the equations and check to show that

Question

the system of equations has one solution. Explain how the graphed lines show the
system has one solution.

Answer 1

To identify a solution to a system of equations and show that it has one solution, let's consider a simple example of a system of linear equations.

Step 1: Identify the equations Let's take the following system of equations:

1. \( y = 2x + 3 \)
2. \( y = -x + 1 \)

Step 2: Find the point of intersection (solution) To find the solution where these two lines intersect, we can set the equations equal to each other:

\[ 2x + 3 = -x + 1 \]

Now, solve for \(x\):

\[ 2x + x = 1 - 3 \] \[ 3x = -2 \] \[ x = -\frac{2}{3} \]

Now, substitute \(x\) back into one of the original equations to find \(y\). We'll use the first equation:

\[ y = 2\left(-\frac{2}{3}\right) + 3 \] \[ y = -\frac{4}{3} + 3 \] \[ y = -\frac{4}{3} + \frac{9}{3} \] \[ y = \frac{5}{3} \]

So, the solution to the system of equations is:

\[ \left(x, y\right) = \left(-\frac{2}{3}, \frac{5}{3}\right) \]

Step 3: Verify the solution Now, we can verify this solution by plugging \(x = -\frac{2}{3}\) back into both original equations to see if we get the same \(y\).

1. For \(y = 2x + 3\): \[ y = 2\left(-\frac{2}{3}\right) + 3 = -\frac{4}{3} + 3 = \frac{5}{3} \]
2. For \(y = -x + 1\): \[ y = -\left(-\frac{2}{3}\right) + 1 = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} \]

Both equations yield the same value of \(y\), confirming that our solution is correct.

Step 4: Graphical representation To explain how the graphed lines show the system has one solution, we can plot the two equations on a coordinate plane:

• The line represented by \(y = 2x + 3\) will have a slope of \(2\) and intercept the y-axis at \(3\).
• The line represented by \(y = -x + 1\) will have a slope of \(-1\) and intercept the y-axis at \(1\).

When graphed, the two lines will intersect at the point \(\left(-\frac{2}{3}, \frac{5}{3}\right)\). Because the lines are not parallel (they have different slopes), they will intersect at exactly one point, indicating the system has one unique solution.

Conclusion Thus, the solution is \(\left(-\frac{2}{3}, \frac{5}{3}\right)\), and the graphical representation shows that since the lines intersect at one point, the system of equations has one solution.

Answer 2

Explain how the graphed lines show the
system has one solution.

Answer 3

To explain how the graphed lines show that a system of linear equations has one solution, we can refer to the basic properties of lines in a coordinate plane.

1. Definition of Linear Equations: Each linear equation in a two-dimensional space (like the ones we discussed) represents a straight line. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

2. Intersection of Lines: The solution to a system of equations corresponds to the point(s) where the lines intersect. If two lines intersect at a single point, that point represents the unique solution to the system of equations.

3. Characteristics of the Lines:

   • Different Slopes: In our example, the equations \(y = 2x + 3\) and \(y = -x + 1\) have different slopes (2 and -1, respectively). Lines with different slopes will always intersect at exactly one point unless they are parallel.
   • No Parallel Lines: If the lines were parallel (meaning they had the same slope but different y-intercepts), they would never intersect, indicating that there is no solution (the system is inconsistent). Since the slopes here are different, this confirms that the lines are not parallel.

4. Graphical Representation:

   • Plotting Points: If you were to plot these lines on a graph, you would start by finding points based on their equations. For instance, for the first equation \(y = 2x + 3\), you could find points by plugging in values for \(x\) (like \(0, 1, -1\), etc.). The same applies to the second equation \(y = -x + 1\).
   • Visualizing the Intersection: Once both lines are drawn on the same coordinate plane, you would visually observe the point at which they cross.

5. Identifying the Unique Solution: The specific point of intersection, which in our case is \(\left(-\frac{2}{3}, \frac{5}{3}\right)\), is where the x and y values satisfy both equations simultaneously. Since this is the only point where the two lines cross, it indicates that there is one and only one solution to the system.

6. Conclusion: By examining the graph, not only can you see the point of intersection indicating the solution, but you can also conclude that:

   • There is exactly one solution since the lines are not parallel.
   • The intersection point corresponds to values \(x\) and \(y\) that satisfy both equations.

Thus, the graphical representation of the lines clearly shows that the system has one solution, as evidenced by the unique point where the two lines intersect.