Engineering Analyses
Methods and Models
ENM 503
Block 2 Lesson 6 Problem Exercises
Methods of Linear Systems
Solutions
1. Find the linear equation in the form y = a + mx
a. Of a line passing through the points (6, -3) and (3,7)
b. Of a line passing through the point (6, -5) and having a slope of -1/4
c. Of a line having a vertical intercept of 10 and a horizontal intercept of 5
ans. a. y = 17 – (10/3)x
b. y = -14/4 – (1/4)x
c. y = 10 – 2x
2. Express the line that has a y-intercept of -4 and x-intercept of 5 in
a. Intercept form: x/5 + y/-4 = 1
b. Slope-intercept form: y = -4 + (4/5)x
c. Point-slope form: y - 4 = (4/5) (x –10)
d. General linear equation form (Ax + By + C = 0): 4x – 5y -20 = 0
3. Solve the following system of equations by (a) substitution and (b) row operations:
a.
2x - 3y + 2z = 21
x + 4y - z = 1
-x + 2y + z = 17
ans.
b.
z=13, y=3, x=2
x + y – 2z = 7
x + y - z = 4
2x – 3y + z = -4
ans. x = 2/5, y = 3/5, z = -3
4. Solve the following system of equations using Excel solver and Eq Solver:
.2x1
.3x1
.25x1
.6x1
.8x1
-.7x1
x1
7237.6
+
+
+
+
x2
-4661
.3x2
.33x2
.3x2
.6x2
2x2
.1x2
–
x3 + .4x4
– .25x3 + .14x4
+
3x3 - .1x4
– .2x3 - .4x4
–
x3 + 44x4
+ .2x3 + .4x4
x3
4618.9
x4
-0.344
x5
-10781
- .5x5 + .1x6
+ .25x5 - .1x6
+ .76x5 +
x6
+ .3x5 - .5x6
+
x5 - .1x6
.5x5 + .1x6
x6
-5907
=
=
=
=
=
=
230
450
167
342
287
191
b
-9493 target
Corrected version: with second equation rewritten as:
.3x1
+ .33x2 – .25x3 + .14x4
x1
x2
x3
-756.8
1204.2
-399.4
x4
+ .25x5 - .1x6 = 450
x5
x6
46.683 898.91 514.66
5. Find the intersection of the following sets: A = {x| 3x – 15 ≤ 210} , B = {x| -x + 25 ≤
230}, C = {x| 3x –32 ≤ 134}, D = {x| 4x + 432 ≥ 225}, E = {x| -5x + 258 ≥ 13}
answer: -51.75 ≤ x ≤ 49
6. Graph the feasible region: x1 ≥ 0, x2 ≥ 0, x1 ≤ 80, x2 ≤ 50,
x1 + x2 ≥ 20, 15x1 – 30x2 ≤ 225
Engineering Analyses
Methods and Models
ENM 503
x1 + 3x2 ≤ 200, -x + y ≤ 40
6. feasible region (not to scale):
y
(10,50)
(40,50)
50
(80,40)
40
(80,32.5)
20
x
10
20
80
7. y = x + 1
8. The Big’nSmall Company manufactures two types of stools requiring the parts shown below:
Type
Legs
1
4
2
3
Available units 256
4x + 3y = 256
x + y = 40
Seat
Nbr
produced
1
1
72
40
32