Worksheet: Consistency and Dependency of Linear Systems

In this worksheet, we will practice determining the number of solutions for a system of linear equations and whether each system is consistent, inconsistent, or dependent.

Q1:

How many solutions are there to the simultaneous equations 𝑥+7𝑦=20 and 2𝑥+14𝑦=40?

  • Atwo solutions
  • Ban infinite number of solutions
  • Czero
  • Da unique solution

Q2:

If the simultaneous equations 6𝑥+7𝑦=2 and 18𝑥+𝑘𝑦=6 have an infinite number of solutions, what is the value of 𝑘?

Q3:

Your friend is thinking of two numbers. She tells you that the numbers add up to 104. Also, two times the first number plus two times the second number is 208. Is it possible to work out the numbers your friend chose?

  • AYes, the system of equations is independent.
  • BNo, the system of equations is inconsistent.
  • CNo, there are infinitely many solutions.

Q4:

Is there a solution to the system of equations 3𝑥+4𝑦=10,3𝑥+4𝑦=12? If yes, what is it?

  • AYes, 𝑥=1,𝑦=2
  • BNo
  • CYes, 𝑥=3,𝑦=4
  • DYes, 𝑥=2,𝑦=1
  • EYes, 𝑥=4,𝑦=3

Q5:

A food drive collected two different types of canned goods: green beans and kidney beans. Overall, they collected 350 cans whose total weight was 348 lb12 oz. If you know that the green bean cans weigh 2 oz less than the kidney bean cans, is it possible to find how many of each can was donated?

  • ANo, there are multiple solutions.
  • BNo, the system is inconsistent.
  • CYes, the system is independent.

Q6:

What is the value of 𝑘 for which the simultaneous equations 𝑥+2𝑦=1 and 5𝑥+𝑘𝑦=5 do not have a unique solution?

Q7:

Admission into an amusement park for 4 children and 2 adults is $116.90. For 6 children and 3 adults, the total price is $175.35. Assuming the price for children and adults is different, what type of system of equations does the scenario create?

  • Adependent system of equations
  • Bindependent system of equations
  • Cinconsistent system of equations

Q8:

What is true, in two dimensions, for two linear equations that simultaneously have infinitely many solutions?

  • ABoth equations represent parallel lines.
  • BBoth equations represent perpendicular lines.
  • CBoth equations represent horizontal lines.
  • DBoth equations represent the same line.
  • EBoth equations represent intersecting lines.

Q9:

The simultaneous equations represented by the straight lines 𝐿 and 𝐿 have an infinite number of solutions. Which of the following describes the relationship between the lines 𝐿 and 𝐿?

  • AThey are coincident.
  • BThey are perpendicular.
  • CThey intersect in one point.
  • DThey are parallel.

Q10:

What property, in two dimensions, is true for two lines represented by two linear equations that do NOT simultaneously have a solution?

  • AThey are perpendicular.
  • BThey are skew.
  • CThey are neither parallel nor perpendicular.
  • DThey are parallel.
  • EThey are coincident.

Q11:

Suppose a system of equations has fewer equations than variables. Must such a system be consistent?

  • ANo
  • BYes

Q12:

If a system of equations has more equations than variables, can it have a solution?

  • Ayes
  • Bno

Q13:

Find the number of solutions to the system of equations

𝑦+𝑥=3,𝑦2𝑥=5.

  • AOne
  • BNone
  • CInfinitely many

Q14:

Find the number of solutions to the system of equations 3𝑦=93𝑥,3𝑦+3𝑥=15.

  • AInfinitely many
  • BNone
  • COne

Q15:

Is there a solution to the simultaneous equations 4𝑥+4𝑦=20,8𝑥+8𝑦=2? If yes, what is it?

  • Ayes, 𝑥=2,𝑦=3
  • Byes, 𝑥=4,𝑦=1
  • Cyes, 𝑥=1,𝑦=1
  • Dno
  • Eyes, 𝑥=1,𝑦=4

Q16:

How many solutions are there to the simultaneous equations 𝑥+9𝑦=11 and 6𝑥+54𝑦=21?

  • Atwo solutions
  • Bzero
  • Can infinite number of solutions
  • Da unique solution

Q17:

How many solutions are there to the simultaneous equations 𝑥+5𝑦=25 and 8𝑥+40𝑦=200?

  • Atwo solutions
  • Ban infinite number of solutions
  • Czero
  • Da unique solution

Q18:

How many solutions are there to the simultaneous equations 𝑥+𝑦=4 and 𝑥𝑦=6?

  • Atwo solutions
  • Ba unique solution
  • Can infinite number of solutions
  • Dzero

Q19:

How many solutions are there to the simultaneous equations 𝑥+6𝑦=16 and 4𝑥+24𝑦=23?

  • Atwo solutions
  • Bzero
  • Can infinite number of solutions
  • Da unique solution

Q20:

How many solutions are there to the simultaneous equations 𝑥+6𝑦=4 and 5𝑥+30𝑦=20?

  • Atwo solutions
  • Ban infinite number of solutions
  • Czero
  • Da unique solution

Q21:

How many solutions are there to the simultaneous equations 𝑥+𝑦=7 and 𝑥𝑦=5?

  • Atwo solutions
  • Ba unique solution
  • Can infinite number of solutions
  • Dzero

Q22:

How many solutions are there to the simultaneous equations 𝑥+7𝑦=18 and 5𝑥+35𝑦=90?

  • Atwo solutions
  • Ban infinite number of solutions
  • Czero
  • Da unique solution

Q23:

How many solutions are there to the simultaneous equations 𝑥+𝑦=16 and 𝑥𝑦=6?

  • Atwo solutions
  • Ba unique solution
  • Can infinite number of solutions
  • Dzero

Q24:

How many solutions can a linear equation with variable 𝑥 have? Pick all possible answers.

  1. 0 solutions
  2. 1 solution
  3. more than 3 solutions
  4. 2 solutions
  • Aa, b, c
  • Ba only
  • Cb, c
  • Da, b
  • Eb only

Q25:

Determine whether the following statement is sometimes, always, or never true: Equations like 𝑎+10=11 and 16𝑚=3 have only one solution.

  • Aalways
  • Bsometimes
  • Cnever

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