Clarification of Answer by
16 Jun 2004 00:01 PDT
Sorry, there is a slight typo in my answer to question 3. I forgot
the decimal point; the answer is C) $8.00, not eight hundred dollars.
Here are some step-by-step solutions.
1. First, isolate x in each equation:
y = 9 ? 2x => 2x = 9 - y
2x ? 5y = 15 => 2x = 15 + 5y
Now 2x = 2x, so we can eliminate that variable:
9 - y = 15 + 5y and now solve for y
6y = -6
-> y = -1
2. We use essentially the same method: isolate a variable and eliminate it
3x + 2y = 3 => y = -1.5x + 1.5
63x ? 36y = 11 => y = 63/36x - 11/36 (after dividing by -36 and moving the x)
63/36x - 11/36 = -1.5x + 1.5
117/36x = 65/36
x = 65/117 = 5/9
let a = the price an adult pays
let c = the price a child pays
Beth and her two children, ages 3 and 4, paid $15...
this means that an adult ticket and 2 child tickets cost 15
so: a + 2c = 15
Jamie and John Jacobs and their 2-year-old daughter got in for $19.50
so: 2a + c = 19.5
Solving these two equations using the same method as above (isolate
and eliminate c)
we get a = 8
So the adult pays $8.
4. Solving for y means we should isolate and eliminate x
2x + y = 3 => 2x = -y + 3
2x + y = ?1 => 2x = -y - 1
-y + 3 = -y - 1 add y to both sides and we get
3 = -1
this is clearly impossible, so that means no pair of numbers x and y
could satisfy the system of equations. This means it is inconsistent.
2x + 3y = 1 => 3y = -2x + 1
5x + 3y = 16 => 3y = -5x + 16
2x - 1 = 5x - 16
3x = 15
-> x = 5
6 to 10 is essentially more of the same, isolating and eliminating
variables. If you would like some help with the steps to these, just
ask and I'll be glad to provide it.
11. An inconsistent system is one with no solutions. In terms of two
lines, it would be two lines which do not cross ever. This means the
lines are parallel.
12. Notice that 4x ? 24y = 32 is just four times x ? 6y = 8 so
really, these describe the same line. That means this system has
infinitely many solutions, not no solutions. False.
The horizontal line running through (3, -2) is y = -2. Solving y = -2
and 2x ? 3y = 6
we get 2x ? 3(-2) = 6 => 3x + 6 = 6 => x = 0
so the point is (0,-2)
14. If two lines intersect at a single point,
- they are not dependent, since dependent lines intersect at
infinitely many points
- they are not inconsistent, since inconsistent lines do not intersect
- to be honest, I have no idea what 'systematic' could mean
but they definitely are independent, as independent lines intersect at one point.