The point (-23, 0) lies on the line -2x + 8y = 46. Determine if this point is a solution of the system
I'm going to show you more than you were asked for, and if you'll pay
attention all the way to the end you'll understand what you need to know
about this subject:
-2x + 8y = 46
x – 2y = -9
The graph of the red equation is the red line graphed below
and the graph of the green equation is the green line graphed below. The
point (-23,0) is circled below and we see that it really is a point on the
red line, but it is NOT a point on the green line.
That's the way to show that (-23,0) is not a solution to the red equation
graphically.
The problem states that the ordered pair is a solution to the
red equation above. The way they showed that algebraically was to
substitute (x,y) = (-23,0) into the red equation.
-2x + 8y = 46
-2(-23) + 8(0) = 46
-2(-23) + 0 = 46
46 = 46
That is true, and that is why they know that (-23,0) lies on the
red line.
To show algebraically that it is not a solution to the system,
we substitute (x,y) = (-23,0) into the green equation.
x – 2y = -9
(-23) – 2(0) = -9
(-23) – 0 = -9
-23 = -9
That's false, so that proves algebraically both that
The ordered pair (-23,0) is not a solution to the green equation
and that the point (-23,0) represents, is not a solution to the
system.
-----------------------
If the problem had been:
Show that the point (5, 7) lies on both the line whose equation
is the red equation below and and also the line whose equation is the
green equation. In other words, show that this point is a solution of
the system:
-2x + 8y = 46
x – 2y = -9
The graph would have been this:
with the point (5,7) circled. And we see that (5,7) is on
both lines, for it is the point where the two lines intersect,
so it is a solution to the system. To show it,
1. Substitute (x,y) = (5,7) it in the red equation:
-2x + 8y = 46
-2(5) + 8(7) = 46
-10 + 56 = 46
46 = 46
That's true. But we have to show that it is also a solution to the green equation:
x – 2y = -9
(5) – 2(7) = -9
5 – 14 = -9
-9 = -9
That's true also. So (5,7) satisfied BOTH equations, so it is on both
lines and therefore is a solution to the system consisting on the red
and the green equations.
Edwin