Question

Consider the following three systems of linear equations.

System C
System A
- 4x + 3y = 15 [A1]
5x-2y = -17 (A2]
System B
8x-6y=-30 [31]
Sx-2y=-17 [82]
- 7x21 (C1)
5x-2y=-17 [02]
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.
How do we transform System A into System B?
08
O
Equation (A1) Equation (81)
х
$
?
1 x Equation [A2] - Equation (B2)
O
Equation (A1) + Equation (A2) Equation [82]
IX Equation (A2) + Equation (A1) - Equation (81)
How do we transform System B into System C?
O
x Equation [81] - Equation (C1]
x Equation (B2) - Equation (C2]
o
x Equation (81) + Equation [82] - Equation (C2]
IX Equation (B2) + Equation (B1Equation (C1]

Answer 1

To transform System A into System B, we perform the transformation: Equation (A1) + Equation (A2) → Equation [82].

To transform System B into System C, the transformation required is: Equation (B2) + Equation (B1) → Equation (C1).

Explanation:

For transforming System A into System B, let's consider the given equations:

- Equation [A1]: -4x + 3y = 15

- Equation [A2]: 5x - 2y = -17

To achieve System B from System A, we apply the operation of adding Equation (A1) to Equation (A2). By doing so, we obtain Equation [82]:

`\[ \text{Equation [A1]} + \text{Equation [A2]} \rightarrow \text{Equation [82]} \]`

(-4x + 3y) + (5x - 2y) = 15 - 17

x + y = -2

To transform System B into System C:

- Equation [31]: 8x - 6y = -30

- Equation [82]: x + y = -2

The necessary transformation involves adding Equation (B2) to Equation (B1) to obtain Equation [C1]:

`\[ \text{Equation (B2)} + \text{Equation (B1)} \rightarrow \text{Equation [C1]} \]`

(x + y) + (8x - 6y) = -2 - 30

9x - 5y = -32

These transformations are fundamental in solving systems of linear equations by manipulating equations to eliminate variables and ultimately find the solution set that satisfies all the given equations simultaneously.