Answer :
To solve the system of equations, we'll go through the following steps:
1. Write down the system of equations:
[tex]\[ \begin{cases} 10x = 5y + 5 \\ y = 2x - 1 \end{cases} \][/tex]
2. Express the first equation in terms of [tex]\(y\)[/tex]:
[tex]\[ 10x = 5y + 5 \][/tex]
Subtract 5 from both sides:
[tex]\[ 10x - 5 = 5y \][/tex]
Divide both sides by 5:
[tex]\[ 2x - 1 = y \][/tex]
3. Notice that the resulting equation [tex]\(y = 2x - 1\)[/tex] is exactly the same as the second equation in the system.
This indicates that the equations are not independent; instead, they represent the same line.
4. Interpret the result:
Since both equations represent the same line, every point [tex]\((x, y)\)[/tex] that satisfies [tex]\(y = 2x - 1\)[/tex] is a solution to the system. Therefore, there are an infinite number of solutions.
In conclusion, the statement "There are an infinite number of solutions" is correct. This occurs because the two equations are essentially the same, implying that any [tex]\((x, y)\)[/tex] pair that satisfies one equation will satisfy the other.
1. Write down the system of equations:
[tex]\[ \begin{cases} 10x = 5y + 5 \\ y = 2x - 1 \end{cases} \][/tex]
2. Express the first equation in terms of [tex]\(y\)[/tex]:
[tex]\[ 10x = 5y + 5 \][/tex]
Subtract 5 from both sides:
[tex]\[ 10x - 5 = 5y \][/tex]
Divide both sides by 5:
[tex]\[ 2x - 1 = y \][/tex]
3. Notice that the resulting equation [tex]\(y = 2x - 1\)[/tex] is exactly the same as the second equation in the system.
This indicates that the equations are not independent; instead, they represent the same line.
4. Interpret the result:
Since both equations represent the same line, every point [tex]\((x, y)\)[/tex] that satisfies [tex]\(y = 2x - 1\)[/tex] is a solution to the system. Therefore, there are an infinite number of solutions.
In conclusion, the statement "There are an infinite number of solutions" is correct. This occurs because the two equations are essentially the same, implying that any [tex]\((x, y)\)[/tex] pair that satisfies one equation will satisfy the other.
