To solve the system of equations by graphing, we will first examine the two equations:
- \( y = 2x - 3 \)
- \( y = -\frac{2}{3}x + 5 \)
Step 1: Graph the equations
For the first equation \( y = 2x - 3 \):
- The y-intercept is at (0, -3).
- The slope is 2, which means for every increase of 1 in x, y increases by 2.
- Starting from the y-intercept (0, -3):
- When \( x = 1 \), \( y = 2(1) - 3 = -1 \), so we plot (1, -1).
- When \( x = 2 \), \( y = 2(2) - 3 = 1 \), so we plot (2, 1).
Connecting these points gives a straight line that crosses the y-axis at (0, -3).
For the second equation \( y = -\frac{2}{3}x + 5 \):
- The y-intercept is at (0, 5).
- The slope is -\(\frac{2}{3}\), which indicates that for every increase of 3 in x, y decreases by 2.
- Starting from the y-intercept (0, 5):
- When \( x = 3 \), \( y = -\frac{2}{3}(3) + 5 = 3 \), so we plot (3, 3).
- When \( x = 6 \), \( y = -\frac{2}{3}(6) + 5 = 1 \), so we plot (6, 1).
Connecting these points gives a straight line that crosses the y-axis at (0, 5).
Step 2: Analyze the graph
When you graph these two lines on a coordinate plane, you will observe that they:
- The line for \( y = 2x - 3 \) slopes upwards.
- The line for \( y = -\frac{2}{3}x + 5 \) slopes downwards.
- Both lines intersect at a certain point.
Step 3: Finding the intersection point
To find the solution to the system, we can either find the intersection graphically or solve the equations algebraically. However, since we are describing the graph, let's focus on the graphical solution.
Description of the Graph
- The first line (increasing) and the second line (decreasing) will intersect at one specific point where the two equations are equal.
- The intersection point indicates the solution to the system of equations. In graphical terms, this is the point (x, y) where both lines meet.
Conclusion
In conclusion, the graph will show two lines intersecting at one point. This indicates that the system of linear equations has a unique solution. The intersection point's coordinates represent the values of \( x \) and \( y \) that satisfy both equations simultaneously.
If you were to calculate the exact coordinates of the intersection point, you could set the equations equal to each other:
\[ 2x - 3 = -\frac{2}{3}x + 5 \]
By solving this equation, you would find the specific coordinates of the intersection. However, based solely on the graph, one can conclude that a unique solution exists where the two lines intersect.