Systems of equations and inequalities

High School Math 1 · Unit 6 · linear-systems · Teacher edition

linear-systems

Name ______________________________Date ______________

1.
Solve the system by substitution and enter $x$ :

$y = 2x - 1, \qquad 3x + y = 14$
Answer: ______________
2.
Solve the system by elimination and enter $y$ :

$3x + 2y = 16, \qquad 5x - 4y = 1$
Answer: ______________
3.
A school play sells $120$ tickets. Adult tickets cost $8 and student tickets cost $5, and the total collected is $780. How many adult tickets were sold?
Answer: ______________
4.
How many solutions does this system have?

$2x - 4y = 8, \qquad x - 2y = 6$
1. (A)
  exactly one solution
2. (B)
  no solution
3. (C)
  infinitely many solutions
4. (D)
  exactly two solutions
5.
Three pens and two notebooks cost $11. Two pens and three notebooks cost $14. What is the cost of one pen and one notebook together?
1. (A)
  $1
2. (B)
  $3
3. (C)
  $4
4. (D)
  $4.50
5. (E)
  $5

Answer key — Systems of equations and inequalities

1.
3linear-systems-01
$3x + 2x - 1 = 14 \Rightarrow 5x = 15 \Rightarrow x = \mathbf{3}$ (and $y = 5$ ). Check: $9 + 5 = 14$ . ✓
2.
7/2linear-systems-02
$2 \times (3x + 2y = 16)$ gives $6x + 4y = 32$ . Adding $5x - 4y = 1$ : $11x = 33$ , $x = 3$ . Then $9 + 2y = 16 \Rightarrow y = \mathbf{\dfrac{7}{2}}$ . Check: $15 - 14 = 1$ . ✓
3.
60linear-systems-03
$8a + 600 - 5a = 780 \Rightarrow 3a = 180 \Rightarrow a = \mathbf{60}$ . Then $s = 60$ ; check $8(60) + 5(60) = 480 + 300 = 780$ . ✓
4.
(B)
no solution
linear-systems-04
The first equation is $x - 2y = 4$ ; the second is $x - 2y = 6$ . Same left side, different constants: the lines are parallel and distinct, so there is no solution. (“Infinitely many” would need the constants to match too; two lines can never cross exactly twice.)
5.
(E)
$5
linear-systems-05
Adding: $5p + 5n = 25$ , so $p + n = \mathbf{5}$ dollars. (Solving fully gives $p = 1$ , $n = 4$ : $1 and $4 are the separate prices, not the sum.)