To solve the system of equations graphically, we will plot the two equations on the same set of axes.
-
Equation 1: \(y = -x + 4\)
- This line has a y-intercept of 4 (0, 4).
- The slope is -1, meaning for every unit you move right (increasing x), you move down 1 unit (decreasing y).
- Another point can be found by picking x = 2:
- \(y = -2 + 4 = 2\) → (2, 2)
Thus, we have two points for the line \(y = -x + 4\): (0, 4) and (2, 2).
-
Equation 2: \(y = 2x - 5\)
- This line has a y-intercept of -5 (0, -5).
- The slope is 2, meaning for every unit you move right, you move up 2 units.
- Picking x = 2:
- \(y = 2(2) - 5 = 4 - 5 = -1\) → (2, -1)
Thus, we have two points for the line \(y = 2x - 5\): (0, -5) and (2, -1).
Graph Representation:
- Line 1 connects (0, 4) and (2, 2).
- Line 2 connects (0, -5) and (2, -1).
Finding the Intersection:
To find the point of intersection of these two lines, set the equations equal to each other: \[ -x + 4 = 2x - 5 \]
Combine like terms: \[ 4 + 5 = 3x \] \[ 9 = 3x \] \[ x = 3 \]
Substituting \(x = 3\) back into either original equation (let’s use the first equation): \[ y = -3 + 4 = 1 \]
Conclusion:
The point of intersection, which is the solution to the system of equations, is: \[ (3, 1) \]
Graphically:
You would click to plot the two lines based on the points we calculated, then identify the intersection point at (3, 1) as the solution to the system.
1 answer