Solving Systems of Linear Equations and Inequalities

Solving Systems of Linear Equations and Inequalities

By Victoria Colbert

Open Ended

OPENING ASSESSMENT:

Tell whether each equation has one, zero, or infinitely many solutions.

1) 5(x - 3) + 6 = 5x - 9 ______________

2) 5(x - 3) + 6 = 5x -10 ______________

3) 5(x - 3) + 6 = 4x + 3 ______________

Systems of Equations

​The graphing method

What is a system of Linear Equations?

A system of linear equations is 2 or more linear equations with the same variables.

One Solution

• The two lines intersect (touch) at ONE POINT

• Your answer should be an ordered pair (x, y)

No Solution

• The two lines NEVER intersect (touch)

• 
  

Infinitely Many Solution

• The equations are on SAME LINE Your answer should be an ordered pair (x, y)

Multiple Choice

^{What is the solution?} 

• 1
• -2
• (1, 2)
• (1, -1)

Multiple Choice

Identify the solution to the system using the graph.

• (0,0)
• (1,1)
• (2,2)
• (3,3)

Solving Systems of Linear Equations by Substitution

Step #1 Isolate a variable in one of the equations to get an expression

Solving Systems

Steps #2 Substitute that expression (from step #1) into the other equation.

Solving Systems

Step #3 Solve for the intersection point (x , y).

Solving Systems

step #4 write your answer as an order pair (x, y).

Multiple Choice

Solve the following system:

$$y=-2$$  

$$4x-3y=18$$  

• $$\left(3,\ -2\right)$$  

• $$\left(3,\ 2\right)$$  

• $$\left(-2,3\right)$$  

• $$\left(2,\ 3\right)$$  

Multiple Choice

Solve the following systems of equations using substitution:

x = 6

y = 2x - 3

• (6, 6)

• (6, 9)

• (9, 6)

• (9, 9)

Multiple Choice

Solve this system by substitution.

• (1, -4)

• (1, 4)

• (-4, -7)

• (-7, -4)

Solving Systems of Linear Equations using elimination

Solving Linear Systems by Elimination

Sometimes we can add equations that have two variables to obtain a new equation that only has one variable. In this sense, we are ELIMINATING a variable in order to solve the system

Steps:

1. Add the equations to eliminate one variable
2. Solve the new equation
3. Substitute value found in step 2 into either original equation to find the value of the other variable

Multiple Choice

Now, let's find the solution to the system.

2x + 3y = 12

4x - 7y = - 54

• (-3, 2)

• (-3, -2)

• (6, -3)

• (-3, 6)

Multiple Choice

Solve the following system:
3x + 2y = 16
7x + y = 19

• (-2,5)
• (-2,-5)
• (2,-5)
• (2,5)

Multiple Choice

Using the elimination method, determine the solution to the system below.

x - y = 11

2x + y = 19

• x = 10, y = -1

• x = -1, y = 10

• x = 3, y = -4

• x = 6, y = 7

Systems of Linear Inequalities

Multiple Choice

Which of the following inequalities matches the given graph?

• y ≥ 2/5x + 1
• y ≥ -2/5x + 1
• y≤  2/5x + 1
• y ≥ -2/5x - 1

Multiple Choice

Which point below is NOT part of the solution set?

• (0, 0)
• (-10, 20)
• (3, 5)
• (0, -2)

Multiple Choice

Which system of inequalities matches the graph?

• y > 2x - 4
  y < 2
• y < 2x - 4
  y < 2
• y > 2x - 4
  y <  2 
• y < 2x - 4
  y <   -2

Multiple Choice

Which system of linear inequalities is represented by the graph?

• y≤2x-1
  y>-2x+3
• y>2x-1
  y≤-2x+3
• y<2x-1
  y≥-2x+3
• y≤2x-1
  y>2x+3

Multiple Choice

Is (0,0) a solution to the system?

• Solution
• Not a solution

Multiple Choice

Which point is a solution?

• (0, 6)
• (-3, 4)
• (1, 0)
• (-4, 3)

Multiple Choice

Which of the following is a solution to the systems of inequalities?

• (0, -2)

• (-3, 0)

• (-3, 2)

• (1, 1)

Multiple Choice

Given the system, determine a solution.

• No solution

• Infinite solutions

• (-5, 5)

• (5, -5)

Multiple Choice

Solve the system of inequalities by graphing.

Multiple Choice

Describe the solution set of the system of equations made by the equation $$y=1.5x+4.5$$  and the graphed line.

• No Solution

• Infinitely Many Solutions

• One Solution

  ( $$\frac{17}{6}$$  , $$\frac{1}{4}$$ )

• One Solution

  ( $$-\frac{17}{6}$$  , $$\frac{1}{4}$$ )

Multiple Choice

Kiyo is considering two catering companies for a party. A+ Food charges $35 per person and $75 to setup. Super Cater charges $38 per person with no setup fee. Write a system of equations to represent the charges for catering by each company.

• A+ Food $$y=35x+75$$  

  Super Cater $$y=38x$$  

• A+ Food $$y=75x\ +\ 35$$  

  Super Cater $$y=0x+38$$  

• A+ Food $$y=35x+75x$$  

  Super Cater $$y=38x$$  

• A+ Food $$y=35+75$$  

  Super Cater $$y=38$$  

Multiple Choice

Kiyo is considering two catering companies for a party. A+ Food charges $35 per person and $75 to setup. Super Cater charges $38 per person with no setup fee. How many people would Kiyo have to invite to pay the same price for both companies? What is the price she would pay?

• Kiyo would have to invite 20 guests to the party and would pay $850

• Kiyo would have to invite 25 guests to the party and would pay $950

• Kiyo would have to invite 850 guests to the party and would pay $20

• Kiyo would have to invite 950 guests to the party and would pay $25

Fill in the Blanks

Use substitution to solve the system of equations.

Write the solution as an ordered pair.

$x=4y−8$  

$3x−6y=12$  

Multiple Choice

Carmen and Alicia go to the office supply store to purchase packs of pens and paper. Carmen bought 5 packs of paper and 3 packs of pens for $36.60. Alicia bought 6 packs of paper and 6 packs of pens for $53.40. Write a system of equations to represent the pens and paper purchased.

• $$5x+3y=53.40$$ $$6x+6y=53.40$$

• $$5x+3y=36.60$$ $$6x+6y=36.60$$

• $$5x-3y=36.60$$ $$6x-6y=53.40$$

• $$5x+3y=36.60$$ $$6x+6y=53.40$$

Multiple Choice

Renaldo has a budget of $500 to buy gift boxes for a party. Large boxes cost $65 and small boxes cost $35. Write an inequality that represents the number of each type of gift box that Renaldo can buy.

• $$65x+35y<500$$  

• $$65x+35y=500$$  

• $$65x+35y\le500$$  

• $$65x+35y\ge500$$  

Multiple Choice

Olivia makes and sells bracelets and necklaces. She can make up to 60 pieces per week, but she can only make up to 40 bracelets and 40 necklaces. Write a system of inequalities that shows the combination of bracelets and necklaces that she can make if she wants to sell at least 30 items per week.

• $$x+y\le60$$   $$x\le40$$   $$y\le40$$   $$x+y\ge30$$  

• $$x+y<60$$   $$x\le40$$   $$y\le40$$   $$x+y>30$$  

• $$x+y\le60$$   $$x<40$$   $$y<40$$   $$x+y\ge30$$  

• $$x+y\ge60$$   $$x\le40$$   $$y\le40$$   $$x+y\le30$$  

Multiple Choice

Olivia makes and sells bracelets and necklaces. She can make up to 60 pieces per week, but she can only make up to 40 bracelets and 40 necklaces. She wants to sell at least 30 items per week. If necklaces sell for $80 each and bracelets sell for $5 each, what is the most money she can make in a week?

• The maximum is $3,300 for

  40 necklaces and 20 bracelets.

• The maximum is $1,800 for

  20 necklaces and 40 bracelets.

• The maximum is $1,300 for

  15 necklaces and 20 bracelets.