To solve the system of equations and determine the expression that produces the vector [tex]\(-10i - 7j\)[/tex], we need to examine and compare each equation step-by-step.
Given equations:
[tex]\[6i + 9j = 2a + b - 3c \tag{1}\][/tex]
[tex]\[2j = 2a + b - 3c \tag{2}\][/tex]
[tex]\[-8i + 13j = \tag{3}\][/tex]
[tex]\[-10i - 7j = \tag{4}\][/tex]
Step-by-step approach:
### Step 1: Identify the remaining vectors
Notice that Equations (1), (2), and (3) are related to vector [tex]\((a, b, c)\)[/tex] expressions. We need to understand these expressions to derive Equation (4).
### Step 2: Compare Equation (1) and Equation (2)
Equation (1) and Equation (2) share the right-hand side, but have different left-hand sides:
[tex]\[6i + 9j\][/tex]
[tex]\[2j\][/tex]
We need to subtract (2) from (1) to eliminate [tex]\(2a + b - 3c\)[/tex]:
[tex]\[ (6i + 9j) - (2j) = 2a + b - 3c - (2a + b - 3c) \][/tex]
[tex]\[ 6i + 7j = 0 \][/tex]
[tex]\[ 6i = -7j \][/tex]
[tex]\[ i = -\frac{7}{6} j \][/tex]
This implies that [tex]\(i\)[/tex] and [tex]\(j\)[/tex] coordinates need to maintain this proportion.
### Step 3: Solve for [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex]
From Equation (3):
[tex]\[ -8i + 13j\][/tex]
We don't need specific values of [tex]\(a, b, c\)[/tex] directly because we are comparing the given expressions.
### Step 4: Analyze Equation (4)
The target vector given:
[tex]\[ -10i - 7j \][/tex]
Realize we need to find the appropriate values aligning with:
[tex]\[
\text{Expression matching }(-10i - 7j)
\][/tex]
### Conclusion
Seeing the equations provided and resulting in comparing the vector expression given as the solution to [tex]\(-10i - 7j\)[/tex], we recognize that these agree with their solving components.
The expression producing [tex]\(-10i - 7j\)[/tex]:
[tex]\((-10, -7)\)[/tex]
Thus, our given vector expression must exactly align with [tex]\( -10i - 7j\)[/tex] in systems (resolved via).
Hence, the required resultant expression produces the vector [tex]\(-10i - 7j\)[/tex].
Answer: [tex]\(-10i - 7j = \boxed{ -10, -7 }\)[/tex]
