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In this lesson, we will learn how to use Cramer's rule to solve a system of linear equations.
Q1:
Is Cramer’s rule useful for finding solutions to systems of linear equations in which there is an infinite set of solutions?
Q2:
William is solving simultaneous equations using Cramer’s rule. He writes down Δ = | | − 9 − 8 2 − 2 | | , Δ = | | 9 − 9 − 3 2 | | .
What does he write down for Δ ?
Q3:
Use determinants to solve the system 3 𝑥 + 2 𝑦 − 2 𝑧 − 1 = 0 , 3 𝑥 + 3 𝑦 − 3 𝑧 + 3 = 0 , − 2 𝑥 + 4 𝑦 − 5 𝑧 + 1 = 0 .
Q4:
Use determinants to solve the system 3 𝑦 = − 3 + 5 𝑧 + 5 𝑥 , 5 𝑥 = 4 𝑦 − 3 𝑧 , 2 𝑥 − 2 𝑦 + 2 𝑧 = 2 .
Q5:
Use determinants to solve the system − 8 𝑥 − 4 𝑦 = − 8 , 9 𝑥 − 6 𝑦 = − 9 .
Q6:
Use determinants to solve the system − 9 𝑥 = − 8 + 8 𝑦 , 6 𝑦 = 7 + 3 𝑥 .
Q7:
Use determinants to solve the system − 7 𝑥 + 2 𝑦 + 5 = 0 , 𝑥 + 8 𝑦 − 1 = 0 .
Q8:
Use determinants to solve the system 5 𝑥 = − 2 𝑦 − 5 + 3 𝑧 , − 3 𝑥 − 𝑦 + 1 = 2 𝑧 , 2 𝑦 − 𝑧 = − 5 𝑥 + 3 .
Q9:
Solve the simultaneous equations | | | − 1 𝑧 − 4 𝑦 | | | = 2 3 , | | | 2 𝑦 − 5 𝑥 | | | = 1 3 , | | 3 𝑥 5 𝑧 | | = 5 1 .
Q10:
11 sketchbooks and 15 crayons cost 377 LE, and 15 sketchbooks and 16 crayons cost 474 LE. Use Cramer’s rule to find the price of a sketchbook and the price of a crayon.
Q11:
Use Cramer’s rule to solve the simultaneous equations | | | − 3 𝑦 2 𝑥 | | | = − 9 , | | | 1 𝑧 1 𝑦 | | | = − 1 , | | 4 𝑥 4 𝑧 | | = 2 .
Q12:
Use determinants to solve the system 𝑥 − 5 𝑦 + 3 𝑧 = 5 , 3 𝑥 − 4 𝑦 + 2 𝑧 = − 5 , − 𝑥 + 3 𝑦 − 2 𝑧 = − 5 .
Q13:
True or False: Suppose that 𝑎 𝑥 + 𝑏 𝑦 = 𝑒 , 𝑐 𝑥 + 𝑑 𝑦 = 𝑓 is a system of equations with 𝑎 𝑑 − 𝑏 𝑐 = 1 . The solution vector is 𝑥 𝑦 = 𝑒 𝑏 𝑓 𝑑 . d e t
Q14:
Given that 9 𝑦 + 5 𝑥 = 8 and 1 𝑦 + 1 𝑥 = 1 , determine the values of 𝑥 and 𝑦 .
Q15:
Use determinants to solve the system − 2 𝑥 − 2 𝑦 + 4 𝑧 + 2 = 0 , 5 𝑥 − 4 𝑦 + 2 𝑧 − 5 = 0 , 2 𝑥 + 3 𝑦 + 𝑧 − 2 = 0 .
Q16:
Use determinants to solve the system 5 𝑥 + 𝑦 − 2 = 0 , 9 𝑥 + 4 𝑦 + 1 = 0 .
Q17:
Use determinants to solve the system − 5 𝑥 = − 2 − 2 𝑦 , − 6 𝑦 = − 5 + 𝑥 .
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