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ACT Exam  >  ACT Questions  >  If the system2x – y + 3z = 2x + y + 2z ... Start Learning for Free
If the system
2x – y + 3z = 2
x + y + 2z = 2
5x – y + az = b
Has infinitely many solutions, then the values of a and b, respectively, are
  • a)
    – 8 and 6
  • b)
    8 and 6
  • c)
    – 8 and –6
  • d)
    8 and –6
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If the system2x – y + 3z = 2x + y + 2z = 25x – y + az = bH...
Concept:
Consider the system of m linear equations
a11 x1 + a12 x2 + … + a1n xn = b1
a21 x1 + a22 x2 + … + a2n xn = b2
…
am1 x1 + am2 x2 + … + amn xn = bm
The above equations containing the n unknowns x1, x2, …, xn. To determine whether the above system of equations is consistent or not, we need to find the rank of following matrices.
A is the coefficient matrix and [A|B] is called as augmented matrix of the given system of equations.
We can find the consistency of the given system of equations as follows:
  • If the rank of matrix A is equal to the rank of augmented matrix and it is equal to the number of unknowns, then the system is consistent and there is a unique solution, i.e.
    Rank of A = Rank of augmented matrix = n
  • If the rank of matrix A is equal to the rank of augmented matrix and it is less than the number of unknowns, then the system is consistent and there are an infinite number of solutions.
    Rank of A = Rank of augmented matrix < n
  • If the rank of matrix A is not equal to the rank of the augmented matrix, then the system is inconsistent, and it has no solution.
    Rank of A ≠ Rank of augmented matrix
Calculation:
Given linear system is
2x – y + 3z = 2
x + y + 2z = 2
5x – y + az = b
Then augmented matrix form is written below;
For rank (A) < n = 3
‘a’ must be = 8
For rank [A|B] < 3, b = 6
Therefore a = 8 & b = 6
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If the system2x – y + 3z = 2x + y + 2z = 25x – y + az = bH...
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Directions:Read the passage and choose the best answer to each question.PassagePeople use many different chemicals each day for common household tasks such as cleaning and food preparation.Since the inception of consumer protection laws, chemicals come with toxicity warning labels, directions about proper use, and cautions about the hazards of improper use. Some household chemicals can be quite dangerous, especially when mixed together. One such example is the reaction that occurs when mixing household bleach (NaOCl) with ammonia (NH3). The by-products of the reaction vary depending on the concentrations of the reactants. The following experiments were conducted to determine the levels at which certain by-products resulted from mixing bleach and ammonia.Experiment 1A known by-product of the reaction of bleach and ammonia is chlorine gas (Cl2). Chlorine gas has an intensely disagreeable suffocating odor, and is very poisonous. To determine the quantities of bleach and ammonia that, when mixed together, produce chlorine gas, a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each of the solutions; the amount added was gradually increased for each trial. The amount of chlorine gas produced in each trial was recorded and graphed in Figure 1.Experiment 2Another known by-product of the reaction of bleach and ammonia is nitrogen trichloride (NCl3). Nitrogen trichloride is a yellow, oily, pungent-smelling liquid, often found as a by-product of chemical reactions between nitrogen containing compounds and chlorine. It is highly explosive.To determine the quantities of bleach and ammonia that, when mixed together, produce NCl3, again a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting NCl3 from each mixture was measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each solution; the quantity addedwas gradually increased for each trial. The amount of nitrogen trichloride produced in each trial was recorded in see Table 1.Experiment 3In yet another reaction, bleach and ammonia combined under certain conditions produce a compound known as chloramine. Chloramine (NH2Cl) is a toxic substance commonly used in low concentrations as a disinfectant in municipal water systems as an alternative to chlorination.To determine the mixture of bleach and ammonia at which NH2Cl is produced, a varying amount of ammonia was added to eight different bleach–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NaOCl in 1 kg of water was used in each trial. A certain quantity of NH3 was added to each solution; the quantity of ammonia added was gradually increased for each trial. The amount of chloramine produced in each trial was recorded in Table 2.Q.If a ninth trial were conducted in Experiment 3, adding 1.25 mol of NH3 to the bleach–water solution, the amount of NH2Cl produced would be closest to

Directions:Read the passage and choose the best answer to each question.PassagePeople use many different chemicals each day for common household tasks such as cleaning and food preparation.Since the inception of consumer protection laws, chemicals come with toxicity warning labels, directions about proper use, and cautions about the hazards of improper use. Some household chemicals can be quite dangerous, especially when mixed together. One such example is the reaction that occurs when mixing household bleach (NaOCl) with ammonia (NH3). The by-products of the reaction vary depending on the concentrations of the reactants. The following experiments were conducted to determine the levels at which certain by-products resulted from mixing bleach and ammonia.Experiment 1A known by-product of the reaction of bleach and ammonia is chlorine gas (Cl2). Chlorine gas has an intensely disagreeable suffocating odor, and is very poisonous. To determine the quantities of bleach and ammonia that, when mixed together, produce chlorine gas, a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each of the solutions; the amount added was gradually increased for each trial. The amount of chlorine gas produced in each trial was recorded and graphed in Figure 1.Experiment 2Another known by-product of the reaction of bleach and ammonia is nitrogen trichloride (NCl3). Nitrogen trichloride is a yellow, oily, pungent-smelling liquid, often found as a by-product of chemical reactions between nitrogen containing compounds and chlorine. It is highly explosive.To determine the quantities of bleach and ammonia that, when mixed together, produce NCl3, again a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting NCl3 from each mixture was measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each solution; the quantity addedwas gradually increased for each trial. The amount of nitrogen trichloride produced in each trial was recorded in see Table 1.Experiment 3In yet another reaction, bleach and ammonia combined under certain conditions produce a compound known as chloramine. Chloramine (NH2Cl) is a toxic substance commonly used in low concentrations as a disinfectant in municipal water systems as an alternative to chlorination.To determine the mixture of bleach and ammonia at which NH2Cl is produced, a varying amount of ammonia was added to eight different bleach–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NaOCl in 1 kg of water was used in each trial. A certain quantity of NH3 was added to each solution; the quantity of ammonia added was gradually increased for each trial. The amount of chloramine produced in each trial was recorded in Table 2.Q.In Experiment 2, different quantities of NaOCl were added to the ammonia solution resulting in the production of nitrogen trichloride. The amounts of nitrogen trichloride produced for 3.00, 3.50, and 4.00 mol of NaOCl added were approximately the same. Which of the following best explains why the production of NCl3 was limited, based on this observation and the results of the experiment?

Directions:Read the passage and choose the best answer to each question.PassagePeople use many different chemicals each day for common household tasks such as cleaning and food preparation.Since the inception of consumer protection laws, chemicals come with toxicity warning labels, directions about proper use, and cautions about the hazards of improper use. Some household chemicals can be quite dangerous, especially when mixed together. One such example is the reaction that occurs when mixing household bleach (NaOCl) with ammonia (NH3). The by-products of the reaction vary depending on the concentrations of the reactants. The following experiments were conducted to determine the levels at which certain by-products resulted from mixing bleach and ammonia.Experiment 1A known by-product of the reaction of bleach and ammonia is chlorine gas (Cl2). Chlorine gas has an intensely disagreeable suffocating odor, and is very poisonous. To determine the quantities of bleach and ammonia that, when mixed together, produce chlorine gas, a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each of the solutions; the amount added was gradually increased for each trial. The amount of chlorine gas produced in each trial was recorded and graphed in Figure 1.Experiment 2Another known by-product of the reaction of bleach and ammonia is nitrogen trichloride (NCl3). Nitrogen trichloride is a yellow, oily, pungent-smelling liquid, often found as a by-product of chemical reactions between nitrogen containing compounds and chlorine. It is highly explosive.To determine the quantities of bleach and ammonia that, when mixed together, produce NCl3, again a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting NCl3 from each mixture was measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each solution; the quantity addedwas gradually increased for each trial. The amount of nitrogen trichloride produced in each trial was recorded in see Table 1.Experiment 3In yet another reaction, bleach and ammonia combined under certain conditions produce a compound known as chloramine. Chloramine (NH2Cl) is a toxic substance commonly used in low concentrations as a disinfectant in municipal water systems as an alternative to chlorination.To determine the mixture of bleach and ammonia at which NH2Cl is produced, a varying amount of ammonia was added to eight different bleach–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NaOCl in 1 kg of water was used in each trial. A certain quantity of NH3 was added to each solution; the quantity of ammonia added was gradually increased for each trial. The amount of chloramine produced in each trial was recorded in Table 2.Q.Each of the following is a by-product resulting from mixing bleach and ammonia EXCEPT

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If the system2x – y + 3z = 2x + y + 2z = 25x – y + az = bHas infinitely many solutions, then the values of a and b, respectively, area)– 8 and 6b)8 and 6c)– 8 and –6d)8 and –6Correct answer is option 'B'. Can you explain this answer?
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