Questions: Consider the following three systems of linear equations. System A System B System C -7x - 6y = -21 [A1] -5x - 3y = -6 [A2] 3x = -9 [B1] -5x - 3y = -6 [B2] x = -3 [C1] -5x - 3y = -6 [C2] Answer the questions below. For each, choose the transformation and then fill in the blank with the correct number The arrow (→) means the expression on the left becomes the expression on the right. (a) How do we transform System A into System B? × Equation [A1] → Equation [B1] × Equation [A2] → Equation [B2] × Equation [A1]+ Equation [A2] → Equation [B2] × Equation [A2]+ Equation [A1] → Equation [B1] (b) How do we transform System B into System C? × Equation [B1] → Equation [C1] × Equation [B2] → Equation [C2] × Equation [B1]+ Equation [B2] → Equation [C2] × Equation [B2]+ Equation [B1] → Equation [C1]

Consider the following three systems of linear equations. System A System B System C -7x - 6y = -21 [A1] -5x - 3y = -6 [A2] 3x = -9 [B1] -5x - 3y = -6 [B2] x = -3 [C1] -5x - 3y = -6 [C2] Answer the questions below. For each, choose the transformation and then fill in the blank with the correct number The arrow (→) means the expression on the left becomes the expression on the right. (a) How do we transform System A into System B? × Equation [A1] → Equation [B1] × Equation [A2] → Equation [B2] × Equation [A1]+ Equation [A2] → Equation [B2] × Equation [A2]+ Equation [A1] → Equation [B1] (b) How do we transform System B into System C? × Equation [B1] → Equation [C1] × Equation [B2] → Equation [C2] × Equation [B1]+ Equation [B2] → Equation [C2] × Equation [B2]+ Equation [B1] → Equation [C1]
Transcript text: Consider the following three systems of linear equations. System A System B System C \[ \left\{\begin{array} { c } { - 7 x - 6 y = - 2 1 [ \mathrm { A } 1 ] } \\ { - 5 x - 3 y = - 6 [ \mathrm { A } 2 ] } \end{array} \left\{\begin{array} { c } { 3 x = - 9 \quad [ \mathrm { B } 1 ] } \\ { - 5 x - 3 y = - 6 [ \mathrm { B } 2 ] } \end{array} \left\{\begin{array}{c} x=-3 \quad[\mathrm{C} 1] \\ -5 x-3 y=-6[\mathrm{C} 2] \end{array}\right.\right.\right. \] Answer the questions below. For each, choose the transformation and then fill in the blank with the correct number The arrow $(\rightarrow)$ means the expression on the left becomes the expression on the right. (a) How do we transform System A into System B? $\times$ Equation $[\mathrm{A} 1] \rightarrow$ Equation $[\mathrm{B} 1]$ $\times$ Equation $[\mathrm{A} 2] \rightarrow$ Equation $[\mathrm{B} 2]$ $\times$ Equation $[\mathrm{A} 1]+$ Equation $[\mathrm{A} 2] \rightarrow$ Equation $[\mathrm{B} 2]$ $\square$ $\times$ Equation $[\mathrm{A} 2]+$ Equation $[\mathrm{A} 1] \rightarrow$ Equation $[\mathrm{B} 1]$ (b) How do we transform System B into System C? $\times$ Equation $[\mathrm{B} 1] \rightarrow$ Equation $[\mathrm{C} 1]$ $\times$ Equation $[B 2] \rightarrow$ Equation [C2] $\times$ Equation $[B 1]+$ Equation $[B 2] \rightarrow$ Equation [C2] $\square$ $\times$ Equation $[B 2]+$ Equation $[B 1] \rightarrow$ Equation $[C 1]
failed

Solution

failed
failed

Solution Steps

To solve the given problem, we need to identify the transformations that convert one system of linear equations into another. We will analyze the equations in each system and determine the operations required to achieve the transformations.

(a) Transforming System A into System B
  1. Compare Equation [A1] and Equation [B1] to find the transformation.
  2. Compare Equation [A2] and Equation [B2] to find the transformation.
(b) Transforming System B into System C
  1. Compare Equation [B1] and Equation [C1] to find the transformation.
  2. Compare Equation [B2] and Equation [C2] to find the transformation.
Step 1: Transforming System A into System B

To determine how to transform System A into System B, we analyze the equations:

  • System A: \[ \begin{align_} A_1: & \quad -7x - 6y = -21 \\ A_2: & \quad -5x - 3y = -6 \end{align_} \]

  • System B: \[ \begin{align_} B_1: & \quad 3x = -9 \\ B_2: & \quad -5x - 3y = -6 \end{align_} \]

From the analysis, we find that:

  • \( A_2 \) transforms to \( B_2 \) as they are identical.
  • \( A_1 \) does not transform to \( B_1 \) since \( -7x - 6y \neq 3x \).

Thus, the transformation from System A to System B is: \[ \text{Transform } A_2 \text{ to } B_2 \text{ is valid.} \]

Step 2: Transforming System B into System C

Next, we analyze the transformation from System B to System C:

  • System B: \[ \begin{align_} B_1: & \quad 3x = -9 \\ B_2: & \quad -5x - 3y = -6 \end{align_} \]

  • System C: \[ \begin{align_} C_1: & \quad x = -3 \\ C_2: & \quad -5x - 3y = -6 \end{align_} \]

From the analysis, we find that:

  • \( B_1 \) transforms to \( C_1 \) since \( 3x = -9 \) simplifies to \( x = -3 \).
  • \( B_2 \) is identical to \( C_2 \).

Thus, the transformation from System B to System C is: \[ \text{Transform } B_1 \text{ to } C_1 \text{ is valid.} \]

Final Answer

The transformations are as follows:

  • For part (a): Transform \( A_2 \) to \( B_2 \) is valid.
  • For part (b): Transform \( B_1 \) to \( C_1 \) is valid.

The answers are: \[ \boxed{\text{(a) A2 to B2 is valid; (b) B1 to C1 is valid.}} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Which feature is shared by the villi and microvilli of the small intestines, the cristae of mitochondria, and thylakoids of chloroplasts? energy provider for the organism amplification of surface area site of catabolic metabolism requirement for absorption and secretion site of anabolic metabolism
failedAnswered
Use like bases to solve the exponential equation. n = (1/32)^(3n) * 8 = 2^6 Recall that exponential functions are one-to-one. In the given equation, are the bases of the exponents the same? If no the same base? When the bases are the same, use the one-to-one property to equate the exponents.
failedAnswered
Predict whether the compounds are soluble or insoluble in water. - Insoluble - KNO3 - AgI - NiCO3
failedAnswered
An apple contains 60 kcals. If 36 kcals are from carbohydrate, then how many grams of carbohydrate does this apple contain? 4 g 9 g 36 g 5 g
failedAnswered
Which of the following is a key difference between the slave societies in the South and the societies with slaves in the North? Enslaved Black people in the North typically had less contact with their slaveholders than in the South. In the North, slavery was most often practiced in urban centers. Slavery was only legal in a few Northern colonies. African and African-descended people did not make up the majority of those enslaved in the North.