Answer :
To determine which system of equations also has a solution of [tex]\((2,0)\)[/tex], let's consider each system one by one. We'll verify whether [tex]\((2,0)\)[/tex] satisfies the equations in each system.
1. System 1:
[tex]\[ \begin{array}{l} 6x - 4y = 6 \\ x + 4y = 2 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(6(2) - 4(0) = 12 = 6\)[/tex]. This is false.
- For the second equation: [tex]\(2 + 4(0) = 2 = 2\)[/tex]. This is true.
Since the first equation is false, [tex]\((2,0)\)[/tex] does not satisfy this system.
2. System 2:
[tex]\[ \begin{array}{l} 6x - 4y = 6 \\ 2x + 8y = 2 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(6(2) - 4(0) = 12 = 6\)[/tex]. This is false.
- For the second equation: [tex]\(2(2) + 8(0) = 4 = 2\)[/tex]. This is false.
Since both equations are false, [tex]\((2,0)\)[/tex] does not satisfy this system.
3. System 3:
[tex]\[ \begin{array}{l} x + 4y = 2 \\ 7x = 14 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(2 + 4(0) = 2 = 2\)[/tex]. This is true.
- For the second equation: [tex]\(7(2) = 14 = 14\)[/tex]. This is true.
Since both equations are true, [tex]\((2,0)\)[/tex] satisfies this system.
4. System 4:
[tex]\[ \begin{array}{l} 6x - 4y = 6 \\ 7x = 14 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(6(2) - 4(0) = 12 = 6\)[/tex]. This is false.
- For the second equation: [tex]\(7(2) = 14 = 14\)[/tex]. This is true.
Since the first equation is false, [tex]\((2,0)\)[/tex] does not satisfy this system.
Therefore, the correct system of equations that also has a solution of [tex]\((2,0)\)[/tex] is:
[tex]\[ \begin{array}{l} x + 4y = 2 \\ 7x = 14 \end{array} \][/tex]
1. System 1:
[tex]\[ \begin{array}{l} 6x - 4y = 6 \\ x + 4y = 2 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(6(2) - 4(0) = 12 = 6\)[/tex]. This is false.
- For the second equation: [tex]\(2 + 4(0) = 2 = 2\)[/tex]. This is true.
Since the first equation is false, [tex]\((2,0)\)[/tex] does not satisfy this system.
2. System 2:
[tex]\[ \begin{array}{l} 6x - 4y = 6 \\ 2x + 8y = 2 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(6(2) - 4(0) = 12 = 6\)[/tex]. This is false.
- For the second equation: [tex]\(2(2) + 8(0) = 4 = 2\)[/tex]. This is false.
Since both equations are false, [tex]\((2,0)\)[/tex] does not satisfy this system.
3. System 3:
[tex]\[ \begin{array}{l} x + 4y = 2 \\ 7x = 14 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(2 + 4(0) = 2 = 2\)[/tex]. This is true.
- For the second equation: [tex]\(7(2) = 14 = 14\)[/tex]. This is true.
Since both equations are true, [tex]\((2,0)\)[/tex] satisfies this system.
4. System 4:
[tex]\[ \begin{array}{l} 6x - 4y = 6 \\ 7x = 14 \end{array} \][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into each equation:
- For the first equation: [tex]\(6(2) - 4(0) = 12 = 6\)[/tex]. This is false.
- For the second equation: [tex]\(7(2) = 14 = 14\)[/tex]. This is true.
Since the first equation is false, [tex]\((2,0)\)[/tex] does not satisfy this system.
Therefore, the correct system of equations that also has a solution of [tex]\((2,0)\)[/tex] is:
[tex]\[ \begin{array}{l} x + 4y = 2 \\ 7x = 14 \end{array} \][/tex]
