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Sagot :
To determine which ordered pair [tex]\((x, y)\)[/tex] is a solution to the system of equations:
[tex]\[ 3x + 4y = 18 \][/tex]
[tex]\[ y = \frac{3}{2}x \][/tex]
We will check each option to see if it satisfies both equations.
### Option A: [tex]\((2, 3)\)[/tex]
1. Plug [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex] into the first equation:
[tex]\[ 3(2) + 4(3) = 6 + 12 = 18 \][/tex]
This satisfies the first equation.
2. Check the second equation with [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex]:
[tex]\[ y = \frac{3}{2}(2) = 3 \][/tex]
This satisfies the second equation.
Since [tex]\((2, 3)\)[/tex] satisfies both equations, [tex]\((2, 3)\)[/tex] is a solution to the system.
### Option B: [tex]\((3, 2.25)\)[/tex]
1. Plug [tex]\(x = 3\)[/tex] and [tex]\(y = 2.25\)[/tex] into the first equation:
[tex]\[ 3(3) + 4(2.25) = 9 + 9 = 18 \][/tex]
This satisfies the first equation.
2. Check the second equation with [tex]\(x = 3\)[/tex] and [tex]\(y = 2.25\)[/tex]:
[tex]\[ y = \frac{3}{2}(3) = 4.5 \][/tex]
This does not satisfy the second equation since [tex]\(2.25 \neq 4.5\)[/tex].
Since [tex]\((3, 2.25)\)[/tex] does not satisfy both equations, it is not a solution to the system.
### Option C: [tex]\((4, 1.5)\)[/tex]
1. Plug [tex]\(x = 4\)[/tex] and [tex]\(y = 1.5\)[/tex] into the first equation:
[tex]\[ 3(4) + 4(1.5) = 12 + 6 = 18 \][/tex]
This satisfies the first equation.
2. Check the second equation with [tex]\(x = 4\)[/tex] and [tex]\(y = 1.5\)[/tex]:
[tex]\[ y = \frac{3}{2}(4) = 6 \][/tex]
This does not satisfy the second equation since [tex]\(1.5 \neq 6\)[/tex].
Since [tex]\((4, 1.5)\)[/tex] does not satisfy both equations, it is not a solution to the system.
### Option D: [tex]\((4, 6)\)[/tex]
1. Plug [tex]\(x = 4\)[/tex] and [tex]\(y = 6\)[/tex] into the first equation:
[tex]\[ 3(4) + 4(6) = 12 + 24 = 36 \][/tex]
This does not satisfy the first equation since [tex]\(36 \neq 18\)[/tex].
2. Check the second equation with [tex]\(x = 4\)[/tex] and [tex]\(y = 6\)[/tex]:
[tex]\[ y = \frac{3}{2}(4) = 6 \][/tex]
This satisfies the second equation.
Since [tex]\((4, 6)\)[/tex] does not satisfy both equations, it is not a solution to the system.
### Conclusion
The ordered pair [tex]\((2, 3)\)[/tex] satisfies both equations in the system. Therefore, the solution to the system of equations [tex]\[ 3x + 4y = 18 \][/tex] and [tex]\[ y = \frac{3}{2}x \][/tex] is:
A. [tex]\((2, 3)\)[/tex]
[tex]\[ 3x + 4y = 18 \][/tex]
[tex]\[ y = \frac{3}{2}x \][/tex]
We will check each option to see if it satisfies both equations.
### Option A: [tex]\((2, 3)\)[/tex]
1. Plug [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex] into the first equation:
[tex]\[ 3(2) + 4(3) = 6 + 12 = 18 \][/tex]
This satisfies the first equation.
2. Check the second equation with [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex]:
[tex]\[ y = \frac{3}{2}(2) = 3 \][/tex]
This satisfies the second equation.
Since [tex]\((2, 3)\)[/tex] satisfies both equations, [tex]\((2, 3)\)[/tex] is a solution to the system.
### Option B: [tex]\((3, 2.25)\)[/tex]
1. Plug [tex]\(x = 3\)[/tex] and [tex]\(y = 2.25\)[/tex] into the first equation:
[tex]\[ 3(3) + 4(2.25) = 9 + 9 = 18 \][/tex]
This satisfies the first equation.
2. Check the second equation with [tex]\(x = 3\)[/tex] and [tex]\(y = 2.25\)[/tex]:
[tex]\[ y = \frac{3}{2}(3) = 4.5 \][/tex]
This does not satisfy the second equation since [tex]\(2.25 \neq 4.5\)[/tex].
Since [tex]\((3, 2.25)\)[/tex] does not satisfy both equations, it is not a solution to the system.
### Option C: [tex]\((4, 1.5)\)[/tex]
1. Plug [tex]\(x = 4\)[/tex] and [tex]\(y = 1.5\)[/tex] into the first equation:
[tex]\[ 3(4) + 4(1.5) = 12 + 6 = 18 \][/tex]
This satisfies the first equation.
2. Check the second equation with [tex]\(x = 4\)[/tex] and [tex]\(y = 1.5\)[/tex]:
[tex]\[ y = \frac{3}{2}(4) = 6 \][/tex]
This does not satisfy the second equation since [tex]\(1.5 \neq 6\)[/tex].
Since [tex]\((4, 1.5)\)[/tex] does not satisfy both equations, it is not a solution to the system.
### Option D: [tex]\((4, 6)\)[/tex]
1. Plug [tex]\(x = 4\)[/tex] and [tex]\(y = 6\)[/tex] into the first equation:
[tex]\[ 3(4) + 4(6) = 12 + 24 = 36 \][/tex]
This does not satisfy the first equation since [tex]\(36 \neq 18\)[/tex].
2. Check the second equation with [tex]\(x = 4\)[/tex] and [tex]\(y = 6\)[/tex]:
[tex]\[ y = \frac{3}{2}(4) = 6 \][/tex]
This satisfies the second equation.
Since [tex]\((4, 6)\)[/tex] does not satisfy both equations, it is not a solution to the system.
### Conclusion
The ordered pair [tex]\((2, 3)\)[/tex] satisfies both equations in the system. Therefore, the solution to the system of equations [tex]\[ 3x + 4y = 18 \][/tex] and [tex]\[ y = \frac{3}{2}x \][/tex] is:
A. [tex]\((2, 3)\)[/tex]
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