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To determine which ordered pair is a solution of the system of equations, we need to consider each pair from the options and substitute the values into both equations to verify if they satisfy the system.
Given system of equations:
1. [tex]\(2x - y = -2\)[/tex]
2. [tex]\(\frac{1}{3}y = x\)[/tex]
We will check each ordered pair one by one.
### Option (a) [tex]\( (0, 2) \)[/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 2 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(0) - 2 = -2 \quad \Rightarrow \quad -2 = -2 \quad \text{(True)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(2) = 0 \quad \Rightarrow \quad \frac{2}{3} \neq 0 \quad \text{(False)} \][/tex]
Since one of the equations is not satisfied, [tex]\((0, 2)\)[/tex] is not a solution.
### Option (b) [tex]\((2, 6)\)[/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 6 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(2) - 6 = -2 \quad \Rightarrow \quad 4 - 6 = -2 \quad \Rightarrow \quad -2 = -2 \quad \text{(True)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(6) = 2 \quad \Rightarrow \quad 2 = 2 \quad \text{(True)} \][/tex]
Since both equations are satisfied, [tex]\((2, 6)\)[/tex] is a solution.
### Option (c) [tex]\( (1, 3) \)[/tex]
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(1) - 3 = -2 \quad \Rightarrow \quad 2 - 3 = -2 \quad \Rightarrow \quad -1 \neq -2 \quad \text{(False)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(3) = 1 \quad \Rightarrow \quad 1 = 1 \quad \text{(True)} \][/tex]
Since one of the equations is not satisfied, [tex]\((1, 3)\)[/tex] is not a solution.
### Option (d) [tex]\( (3, 8) \)[/tex]
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 8 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(3) - 8 = -2 \quad \Rightarrow \quad 6 - 8 = -2 \quad \Rightarrow \quad -2 = -2 \quad \text{(True)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(8) = 3 \quad \Rightarrow \quad \frac{8}{3} \neq 3 \quad \text{(False)} \][/tex]
Since one of the equations is not satisfied, [tex]\((3, 8)\)[/tex] is not a solution.
### Conclusion
The ordered pair that satisfies both equations in the system is [tex]\((2, 6)\)[/tex].
Therefore, the correct answer is option (b) [tex]\((2, 6)\)[/tex].
Given system of equations:
1. [tex]\(2x - y = -2\)[/tex]
2. [tex]\(\frac{1}{3}y = x\)[/tex]
We will check each ordered pair one by one.
### Option (a) [tex]\( (0, 2) \)[/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 2 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(0) - 2 = -2 \quad \Rightarrow \quad -2 = -2 \quad \text{(True)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(2) = 0 \quad \Rightarrow \quad \frac{2}{3} \neq 0 \quad \text{(False)} \][/tex]
Since one of the equations is not satisfied, [tex]\((0, 2)\)[/tex] is not a solution.
### Option (b) [tex]\((2, 6)\)[/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 6 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(2) - 6 = -2 \quad \Rightarrow \quad 4 - 6 = -2 \quad \Rightarrow \quad -2 = -2 \quad \text{(True)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(6) = 2 \quad \Rightarrow \quad 2 = 2 \quad \text{(True)} \][/tex]
Since both equations are satisfied, [tex]\((2, 6)\)[/tex] is a solution.
### Option (c) [tex]\( (1, 3) \)[/tex]
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(1) - 3 = -2 \quad \Rightarrow \quad 2 - 3 = -2 \quad \Rightarrow \quad -1 \neq -2 \quad \text{(False)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(3) = 1 \quad \Rightarrow \quad 1 = 1 \quad \text{(True)} \][/tex]
Since one of the equations is not satisfied, [tex]\((1, 3)\)[/tex] is not a solution.
### Option (d) [tex]\( (3, 8) \)[/tex]
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 8 \)[/tex].
1. For [tex]\( 2x - y = -2 \)[/tex]:
[tex]\[ 2(3) - 8 = -2 \quad \Rightarrow \quad 6 - 8 = -2 \quad \Rightarrow \quad -2 = -2 \quad \text{(True)} \][/tex]
2. For [tex]\( \frac{1}{3}y = x \)[/tex]:
[tex]\[ \frac{1}{3}(8) = 3 \quad \Rightarrow \quad \frac{8}{3} \neq 3 \quad \text{(False)} \][/tex]
Since one of the equations is not satisfied, [tex]\((3, 8)\)[/tex] is not a solution.
### Conclusion
The ordered pair that satisfies both equations in the system is [tex]\((2, 6)\)[/tex].
Therefore, the correct answer is option (b) [tex]\((2, 6)\)[/tex].
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