Unit 6 Study Guide - Systems of Linear Equations & Inequalities

Graph this system and determine the number of solutions it has. If it has 1 solution, name it.

y=2x

y=6-x

• One solution

  (2, 4)

• One Solution

  (3, 4)

• No Solution

• Infinitely Many Solutions

Answer explanation

To find the solution, graph the equations y=2x and y=6-x. They intersect at (2, 4), indicating one solution. Thus, the correct answer is one solution at (2, 4).

Tags

CCSS.8.EE.C.8B

Graph this system and determine the number of solutions it has. If it has 1 solution, name it.

y=x-3

y=-2x+9

• One Solution

  (1, 2)

• No Solution

• One Solution

  (4, 1)

• One Solution

  (2, 1)

Answer explanation

To find the intersection of the lines y=x-3 and y=-2x+9, set them equal: x-3 = -2x+9. Solving gives x=4, y=1. Thus, the system has one solution at (4, 1). This matches the correct answer.

Tags

CCSS.8.EE.C.8B

CCSS.HSA.REI.C.6

Graph on graph paper and determine what the solution is:

2y − x = 4
4y − 2x = 8

• (2, 4)

• (4, 2)

• Infinitely Many Solutions

• No Solution

Tags

CCSS.8.EE.C.8B

Use substitution to solve this system of equations:

y=x+8

2x+y=-10

• (2, 4)

• (5, 2)

• (6, 3)

• (-6, 2)

Answer explanation

Substituting y in the second equation: 2x + (x + 8) = -10 simplifies to 3x + 8 = -10. Solving gives x = -6. Substituting x back into y = x + 8 gives y = 2. Thus, the solution is (-6, 2).

Tags

CCSS.8.EE.C.8B

CCSS.HSA.REI.C.6

For the following system of equations:

x = 2y + 1
3y + x = 7

Using the substitution method, which of the following shows the correct first step?

• Replace y in the first equation with (7 − x)/3 → x = 2((7 − x)/3) + 1

• Replace x in the second equation with 2y + 1 → 3y + (2y + 1) = 7

• Add the two equations together → x + (3y + x) = (2y + 1) + 7

• Subtract 2y from both sides of the first equation → x − 2y = 1

Tags

CCSS.8.EE.C.8B

CCSS.HSA.REI.C.6

Corey has 42 of fencing around his garden.

The garden is rectangular in shape, and its length is equal to twice the width minus 3 feet.

Which system could be used to find the length and width of the garden.

Solve by substitution.

• 2w-3=42

  L=2w+2L

  .

  w=9 feet

  L=13 feet

• 2w+2L=42

  L=2w-3

  .

  w=8 feet

  L=13 feet

• 2w-3=42

  L=2w+2L

  .

  w=13 feet

  L=9 feet

• 2w+2L=42

  L=2w-3

  .

  w=13 feet

  L=13 feet

Answer explanation

The correct system is 2w+2L=42 and L=2w-3. This represents the perimeter of the rectangle and the relationship between length and width. Solving gives w=9 feet and L=13 feet.

Tags

CCSS.8.EE.C.8C

Use elimination to solve the system.

6x-4y=6

-6x+3y=0

• (5, 6)

• (-3, -6)

• (1, 0)

• (4, -8)

Answer explanation

To eliminate x, add the equations: (6x - 4y) + (-6x + 3y) = 6 + 0. This simplifies to -y = 6, so y = -6. Substitute y back into one equation to find x: 6x - 4(-6) = 6, giving x = -3. Thus, the solution is (-3, -6).

Tags

CCSS.8.EE.C.8B

CCSS.HSA.REI.C.6

Use elimination to solve this system of equations:

x+y=13

x-y=5

• (9, 4)

• (4, 9)

• (6, 2)

• (2, 6)

Answer explanation

To eliminate y, add the equations: (x+y) + (x-y) = 13 + 5, resulting in 2x = 18, so x = 9. Substitute x back into x+y=13: 9+y=13, giving y=4. Thus, the solution is (9, 4).

Tags

CCSS.8.EE.C.8B

CCSS.HSA.REI.C.6

Use elimination to solve this system of equations:

2x+6y=18

3x+2y=13

• (4, 1)

• (2, 3)

• (1, 4)

• (3, 2)

Answer explanation

To eliminate y, multiply the second equation by 3: 9x + 6y = 39. Subtract the first equation from this: 7x = 21, giving x = 3. Substitute x back to find y: 3(3) + 2y = 13, yielding y = 2. Thus, the solution is (3, 2).

Tags

CCSS.8.EE.C.8B

CCSS.HSA.REI.C.6

Shelly has $300 to shop for jeans and sweaters.

Each pair of jeans cost $65, each sweater costs $34, and she buys 7 items.

Determine the number of pairs of jeans and sweaters Shelly bought.

• 5 jeans and 2 sweaters

• 3 jeans and 4 sweaters

• 4 jeans and 3sweaters

• 2 jeans and 5 sweaters

Answer explanation

Let x be jeans and y be sweaters. We have the equations: 65x + 34y = 300 and x + y = 7. Solving these, we find x = 2 and y = 5. Thus, Shelly bought 2 jeans and 5 sweaters.

Tags

CCSS.8.EE.C.8C