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).

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.

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).

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.

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).

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.

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).

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).

Julie subscribes to a sports magazine and a fashion magazine.

She received 24 issues this year.

The number of fashion issues is 6 less than twice the number of sports issues.

Define the variables, and determine the system of equations fo find the number of issues of each magazine.

• f = number of fashion magazines

  s = number of sports magazines

  f+s=6

  f=2s-24

• f = number of fashion magazines

  s = number of sports magazines

  f+s=24

  f=2s-6

• f = number of fashion magazines

  s = number of sports magazines

  f-s=6

  f=2s-24

• f = number of fashion magazines

  s = number of sports magazines

  f-s=24

  f=2s-6

Answer explanation

Let f = fashion issues and s = sports issues. The total issues are f + s = 24. The fashion issues are 6 less than twice the sports issues: f = 2s - 6. Thus, the correct system is f + s = 24 and f = 2s - 6.

Determine the best method for solving this system. Then, solve it.

3x+5y=7

2x-3y=11

• Substitution

  (4, -1)

• Graphing

  (4, -1)

• Elimination by Subtraction

  (4, -1)

• Elimination by Multiplication

  (4, -1)

Answer explanation

The best method is Elimination by Multiplication. By multiplying the first equation by 2 and the second by 3, we can eliminate y and solve for x. This leads to the solution (4, -1).