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Find the points of intersection of the following lines:
2x - y = 8
4x - 2y = 16
Solution:
Since the given equations are all first degree, then they are linear equations. They are straight lines. We can graph the two lines by getting their slope and y-intercept.
For 2x - y = 8
2x - y = 8
-y = -2x + 8
y = 2x - 8
slope (Δy/Δx), m = 2
y-intercept, b = -8
To trace the graph, plot -8 at the y-axis. This is your first point of the line (0, -8). Next, use the slope to get the second point. From the first point, count 1 unit to the right and then 2 units upward.
For 4x - 2y = 16
4x - 2y = 16
-2y = -4x + 16
y = 2x - 8
slope (Δy/Δx), m = 2
y-intercept, b = -8
To trace the graph, plot -8 at the y-axis. This is your first point of the line (0, -8). Next, use the slope to get the second point. From the first point, count 1 unit to the right and then 2 units upward.
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From the graph, the two lines are coincide to each other because their slopes and y-intercepts are the same which are 2 and -8. The two lines will have an infinite number of points of intersection. When you solve for x and y from the two given equations, their x, y, and constant will be equal to zero. From the two given equations,
2x - y = 8
4x - 2y = 16
Multiply the first equation by 2 and -1 at the second equation. Add the two equations and let's see what will happen to x, y, and constant.
2 (2x - y = 8) 4x - 2y = 16
→
-1 (4x - 2y = 16) -4x + 2y = -16
_______________
0 = 0
Since everything in the equation are all equal to zero, then there's no way that we can solve for x and y. Therefore, the two lines are coincide to each other.