Determine whether each system has a unique solution. If it has a unique solution, find it.
-
{
20
x
+
5
y
=
240
y
=
20
x
left brace . table with 2 rows and 1 column , row1 column 1 , 20 x plus 5 y equals 240 , row2 column 1 , y equals 20 x , end table
-
{
20
x
+
5
y
=
145
30
x
−
5
y
=
125
left brace . table with 2 rows and 1 column , row1 column 1 , 20 x plus 5 y equals 145 , row2 column 1 , 30 x minus 5 y equals 125 , end table
-
{
y
=
2000
−
65
x
y
=
500
+
55
x
left brace . table with 2 rows and 1 column , row1 column 1 , y equals , 2000 , minus 65 x , row2 column 1 , y equals 500 plus 55 x , end table
-
{
y
=
2
3
x
−
3
y
=
−
x
+
7
left brace . table with 2 rows and 1 column , row1 column 1 , y equals , 2 thirds , x minus 3 , row2 column 1 , y equals negative x plus 7 , end table
-
{
3
x
+
2
y
=
10
6
x
+
4
y
=
16
left brace . table with 2 rows and 1 column , row1 column 1 , 3 x plus 2 y equals 10 , row2 column 1 , 6 x plus 4 y equals 16 , end table
-
{
x
+
2
y
+
z
=
4
y
=
x
−
3
z
=
2
x
left brace . table with 3 rows and 1 column , row1 column 1 , x plus 2 y plus z equals 4 , row2 column 1 , y equals x minus 3 , row3 column 1 , z equals 2 x , end table
-
Coordinate Geometry The coordinates (x, y) of a point in a plane are the solution of the system
{
2
x
+
3
y
=
13
5
x
+
7
y
=
31
.
left brace . table with 2 rows and 1 column , row1 column 1 , 2 x plus 3 y equals 13 , row2 column 1 , 5 x plus 7 y equals 31 , end table . . Find the coordinates of the point.
-
Geometry A rectangle is twice as long as it is wide. The perimeter is 840 ft. Find the dimensions of the rectangle.
-
Reasoning Substitute each point
(
−
3
,
5
)
open negative 3 comma 5 close and
(
2
,
−
1
)
open 2 comma negative 1 close into the slope-intercept form of a linear equation to write a system of equations. Then use the system to find the equation of the line containing the two points. Explain your reasoning.
C Challenge
Solve each matrix equation.
-
−
2
[
−
2
0
0
−
1
]
+
[
0
−
3
5
−
4
]
X
+
[
0
−
3
5
−
4
]
=
[
19
−
27
10
−
24
]
negative 2 . matrix with 2 rows and 2 columns , row1 column 1 , negative 2 , column 2 0 , row2 column 1 , 0 , column 2 negative 1 , end matrix . plus . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 negative 3 , row2 column 1 , 5 , column 2 negative 4 , end matrix x plus . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 negative 3 , row2 column 1 , 5 , column 2 negative 4 , end matrix . equals . matrix with 2 rows and 2 columns , row1 column 1 , 19 , column 2 negative 27 , row2 column 1 , 10 , column 2 negative 24 , end matrix
-
[
0
−
6
1
2
]
−
[
5
2
4
3
]
X
−
[
2
−
26
3
−
18
]
=
[
3
25
2
24
]
. matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 negative 6 , row2 column 1 , 1 , column 2 2 , end matrix . minus . matrix with 2 rows and 2 columns , row1 column 1 , 5 , column 2 2 , row2 column 1 , 4 , column 2 3 , end matrix x minus . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 26 , row2 column 1 , 3 , column 2 negative 18 , end matrix . equals . matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 25 , row2 column 1 , 2 , column 2 24 , end matrix
-
[
7
−
5
3
0
1
3
8
4
−
2
]
X
+
[
5
−
9
0
]
=
[
54
−
12
96
]
. matrix with 3 rows and 3 columns , row1 column 1 , 7 , column 2 negative 5 , column 3 3 , row2 column 1 , 0 , column 2 1 , column 3 3 , row3 column 1 , 8 , column 2 4 , column 3 negative 2 , end matrix x plus . matrix with 3 rows and 1 column , row1 column 1 , 5 , row2 column 1 , negative 9 , row3 column 1 , 0 , end matrix . equals . matrix with 3 rows and 1 column , row1 column 1 , 54 , row2 column 1 , negative 12 , row3 column 1 , 96 , end matrix
-
[
−
1
0
2
−
6
−
5
0
1
4
1
]
[
−
4
0
2
0
3
6
0
5
0
]
X
=
[
−
21
10
26
−
54
1
−
15
1
4
−
24
]
. matrix with 3 rows and 3 columns , row1 column 1 , negative 1 , column 2 0 , column 3 2 , row2 column 1 , negative 6 , column 2 negative 5 , column 3 0 , row3 column 1 , 1 , column 2 4 , column 3 1 , end matrix . matrix with 3 rows and 3 columns , row1 column 1 , negative 4 , column 2 0 , column 3 2 , row2 column 1 , 0 , column 2 3 , column 3 6 , row3 column 1 , 0 , column 2 5 , column 3 0 , end matrix x equals . matrix with 3 rows and 3 columns , row1 column 1 , negative 21 , column 2 10 , column 3 26 , row2 column 1 , negative 54 , column 2 1 , column 3 negative 15 , row3 column 1 , 1 , column 2 4 , column 3 negative 24 , end matrix
Open-Ended Complete each system for the given number of solutions.
-
infinitely many
{
x
+
y
=
7
2
x
+
2
y
=
□
left brace . table with 2 rows and 2 columns , row1 column 1 , x plus y , column 2 equals 7 , row2 column 1 , 2 x plus 2 y , column 2 equals white square , end table
-
one solution
{
x
+
y
+
z
=
7
y
+
z
=
□
z
=
□
left brace . table with 3 rows and 2 columns , row1 column 1 , x plus y plus z , column 2 equals 7 , row2 column 1 , y plus z , column 2 equals white square , row3 column 1 , z , column 2 equals white square , end table
-
no solution
{
x
+
y
+
z
=
7
y
+
z
=
□
y
+
z
=
□
left brace . table with 3 rows and 2 columns , row1 column 1 , x plus y plus z , column 2 equals 7 , row2 column 1 , y plus z , column 2 equals white square , row3 column 1 , y plus z , column 2 equals white square , end table
-
Nutrition A caterer combines ingredients to make a paella, a Spanish fiesta dish. The paella weighs 18 lb, costs $29.50, and supplies 850 g of protein.
Paella Nutrition Chart
| Food |
Cost/Ib |
Protein/Ib |
| Chicken |
$1.50 |
100 g |
| Rice |
$.40 |
20 g |
| Shellfish |
$6.00 |
50 g |
- Write a system of three equations to find the weight of each ingredient that the caterer uses.
- Solve the system. How many pounds of each ingredient did she use?
