Lesson: Number of Solutions to a System of Linear Equations

In this lesson, we will learn how to determine the number of solutions for a system of linear equations and whether each system is consistent, inconsistent, or dependent.

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Worksheet: Number of Solutions to a System of Linear Equations • 25 Questions • 3 Videos

Q1:

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

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?

Q4:

Is there a solution to the simultaneous equations If yes, what is it?

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?

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?

Q8:

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

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?

Q10:

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 ?

Q11:

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

Q12:

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

Q13:

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

Q14:

Find the number of solutions to the system of equations

Q15:

Find the number of solutions to the system of equations

Q16:

5 = 5 Γ— 5 4 3 1 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 π‘₯ 𝑦 π‘₯ + 𝑦 .

Which of the following is equivalent to 5 9 ?

Q17:

Is there a solution to the simultaneous equations If yes, what is it?

Q18:

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

Q19:

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

Q20:

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

Q21:

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

Q22:

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

Q23:

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

Q24:

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

Q25:

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

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