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Worksheet: Determining the Number of Solutions a System of Linear Equations Has

Q1:

How many solutions are there to the simultaneous equations π‘₯ + 7 𝑦 = 2 0 and 2 π‘₯ + 1 4 𝑦 = 4 0 ?

  • Aa unique solution
  • Btwo solutions
  • Czero
  • Dan infinite number of solutions

Q2:

How many solutions are there to the simultaneous equations 𝑦 = π‘₯ and 𝑦 + π‘₯ = 1 6 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q3:

How many solutions are there to the simultaneous equations 𝑦 = 8 and 𝑦 βˆ’ π‘₯ = 1 2 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q4:

How many solutions are there to the simultaneous equations 𝑦 = 2 and 𝑦 βˆ’ π‘₯ = 1 9 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q5:

How many solutions are there to the simultaneous equations 𝑦 = 7 and 𝑦 βˆ’ π‘₯ = 1 7 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q6:

How many solutions are there to the simultaneous equations π‘₯ + 9 𝑦 = 1 1 and 6 π‘₯ + 5 4 𝑦 = 2 1 ?

  • Aa unique solution
  • Btwo solutions
  • Can infinite number of solutions
  • Dzero

Q7:

How many solutions are there to the simultaneous equations π‘₯ + 5 𝑦 = 2 5 and 8 π‘₯ + 4 0 𝑦 = 2 0 0 ?

  • Aa unique solution
  • Btwo solutions
  • Czero
  • Dan infinite number of solutions

Q8:

How many solutions are there to the simultaneous equations π‘₯ + 𝑦 = 4 and π‘₯ βˆ’ 𝑦 = 6 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q9:

How many solutions are there to the simultaneous equations π‘₯ + 6 𝑦 = 1 6 and 4 π‘₯ + 2 4 𝑦 = 2 3 ?

  • Aa unique solution
  • Btwo solutions
  • Can infinite number of solutions
  • Dzero

Q10:

How many solutions are there to the simultaneous equations 𝑦 = 7 and 𝑦 βˆ’ π‘₯ = 1 8 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q11:

How many solutions are there to the simultaneous equations π‘₯ + 6 𝑦 = 4 and 5 π‘₯ + 3 0 𝑦 = 2 0 ?

  • Aa unique solution
  • Btwo solutions
  • Czero
  • Dan infinite number of solutions

Q12:

How many solutions are there to the simultaneous equations 𝑦 = 7 and 𝑦 βˆ’ π‘₯ = 2 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q13:

How many solutions are there to the simultaneous equations π‘₯ + 𝑦 = 7 and π‘₯ βˆ’ 𝑦 = 5 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q14:

How many solutions are there to the simultaneous equations π‘₯ + 7 𝑦 = 1 8 and 5 π‘₯ + 3 5 𝑦 = 9 0 ?

  • Aa unique solution
  • Btwo solutions
  • Czero
  • Dan infinite number of solutions

Q15:

How many solutions are there to the simultaneous equations 𝑦 = 3 and 𝑦 βˆ’ π‘₯ = 2 5 ?

  • Azero
  • Btwo solutions
  • Can infinite number of solutions
  • Da unique solution

Q16:

If the simultaneous equations 6 π‘₯ + 7 𝑦 = 2 and 1 8 π‘₯ + π‘˜ 𝑦 = 6 have an infinite number of solutions, what is the value of π‘˜ ?

Q17:

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

  • ANo
  • BYes

Q18:

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 lb 12 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?

  • AYes, the system is independent.
  • BNo, the system is inconsistent.
  • CNo, there are multiple solutions.

Q19:

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

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

Q20:

What is the value of π‘˜ for which the simultaneous equations π‘₯ + 2 𝑦 = 1 and 5 π‘₯ + π‘˜ 𝑦 = 5 do not have a unique solution?

Q21:

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

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

Q22:

Consider the three equations

  1. π‘₯ + 1 = 3 ,
  2. ( π‘₯ + 1 ) = 9  ,
  3. ( π‘₯ + 1 ) = 2 7  .

Which equations have the same solution set?

  • Anone
  • B(A), (B), and (C)
  • C(A) and (B) only
  • D(A) and (C) only
  • E(B) and (C) only

Q23:

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?

  • Aindependent system of equations
  • Binconsistent system of equations
  • Cdependent system of equations

Q24:

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.

Q25:

The simultaneous equations represented by the straight lines 𝐿 1 and 𝐿 2 have an infinite number of solutions. Which of the following describes the relationship between the lines 𝐿 1 and 𝐿 2 ?

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