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