Worksheet: Number of Solutions to a System of Linear Equations

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 π‘₯ + 7 𝑦 = 2 0 and 2 π‘₯ + 1 4 𝑦 = 4 0 ?

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

Q2:

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‘˜ ?

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

Q4:

Is there a solution to the system of equations 3 π‘₯ + 4 𝑦 = 1 0 , 3 π‘₯ + 4 𝑦 = βˆ’ 1 2 ? If yes, what is it?

  • Ayes, π‘₯ = 2 , 𝑦 = 1
  • Bno
  • Cyes, π‘₯ = 3 , 𝑦 = 4
  • Dyes, π‘₯ = 1 , 𝑦 = 2
  • Eyes, π‘₯ = 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 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.

Q6:

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?

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
  • Binconsistent system of equations
  • Cindependent system of equations

Q8:

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

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

Q9:

Consider the three equations

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

Which equations have the same solution set?

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

Q10:

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

Q11:

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

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

Q12:

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

  • AYes
  • BNo

Q13:

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

  • Ano
  • Byes

Q14:

Find the number of solutions to the system of equations

𝑦 + π‘₯ = 3 , 𝑦 βˆ’ 2 π‘₯ = 5 .

  • AInfinitely many
  • BNone
  • COne

Q15:

Find the number of solutions to the system of equations 3 𝑦 = 9 βˆ’ 3 π‘₯ , 3 𝑦 + 3 π‘₯ = 1 5 .

  • AInfinitely many
  • BNone
  • COne

Q16:

5 = 5 Γ— 5 οŠͺ   and 5 Γ— 5 Γ— 5 Γ— 5 = ( 5 Γ— 5 Γ— 5 ) Γ— ( 5 ) .

This is an example of the product rule of exponents π‘Ž Γ— π‘Ž = π‘Ž      .

Which of the following is equivalent to 5  ?

  • A 5 Γ— 5  
  • B 5 Γ— 5  
  • C 5 Γ— 6 οŠͺ 
  • D 5 + 5  
  • E 5 Γ— 5  

Q17:

Is there a solution to the simultaneous equations 4 π‘₯ + 4 𝑦 = 2 0 , 8 π‘₯ + 8 𝑦 = βˆ’ 2 ? If yes, what is it?

  • Ayes, π‘₯ = 1 , 𝑦 = 1
  • Bno
  • Cyes, π‘₯ = 4 , 𝑦 = βˆ’ 1
  • Dyes, π‘₯ = 2 , 𝑦 = 3
  • Eyes, π‘₯ = 1 , 𝑦 = 4

Q18:

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

  • Azero
  • Ban infinite number of solutions
  • Ctwo solutions
  • Da unique solution

Q19:

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
  • Bzero
  • Can infinite number of solutions
  • Dtwo solutions

Q20:

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

Q21:

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
  • Czero
  • Dan infinite number of solutions

Q22:

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

  • Atwo solutions
  • Ba unique solution
  • Czero
  • Dan infinite number of solutions

Q23:

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

  • Azero
  • Ban infinite number of solutions
  • Ca unique solution
  • Dtwo solutions

Q24:

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
  • Bzero
  • Ctwo solutions
  • Dan infinite number of solutions

Q25:

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

  • Azero
  • Ba unique solution
  • Ctwo solutions
  • Dan infinite number of solutions

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