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1. Math 085.03 - Quiz 1 Solutions
(1) Below are two lines that form a linear system.
(a) List all solutions of the system.
Solutions are just the intersection points of the lines. So there is only one, (−2, 3).
(b) Write the system of linear equations that correspond to this system.
We just need to write the equations of the lines. First we’ll calculate the slopes, using
the points I’ve indicated on the diagram:
2
1−3
=−
3 − (−2)
5
1
4−3
= .
4 − (−2)
6
Now we can take the slopes and put them in to the point-slope form of the equation
of a line. I’ll use (−2, 3) for the point, since it is on both lines.
(
(
y − 3 = − 25 (x + 2)
y = − 25 x + 11
5
⇒
y − 3 = 16 (x + 2)
y = 61 x + 10
.
3
(MORE ON BACK)
1
2
(2) Solve(the linear systems below.
x + 2y = 8
(a)
x − 3y = 3
It seems easiest to use elimination. Subtracting the second equation from the first
gives
5y = 5
⇒
y = 1.
Putting this back into the first equation, we get
x + 2(1) = 8
⇒
x = 6.
So the solution is (6, 1).
(
y = 4x + 5
(b)
−8x + 2y = 2
Since the first equation is already solved for y, it is easiest to use substitution. The
second equation becomes
−8x + 2(4x + 5) = 2
−8x + 8x + 10 = 2
10 = 2.
There are no values for x that make this true, so there are no solutions to this system
of equations.