Answer :
To determine the value of [tex]\( x \)[/tex] at which the graphs of the equations intersect, we need to solve the system of equations:
[tex]\[ \begin{cases} 2x - y = 6 \\ 5x + 10y = -10 \end{cases} \][/tex]
Let's go through the solution step by step:
1. Isolate one variable in one of the equations:
Start by isolating [tex]\( y \)[/tex] in the first equation:
[tex]\[ 2x - y = 6 \][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ -y = 6 - 2x \][/tex]
Then multiply by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 6 \][/tex]
2. Substitute the expression for [tex]\( y \)[/tex] into the other equation:
Now substitute [tex]\( y = 2x - 6 \)[/tex] into the second equation:
[tex]\[ 5x + 10y = -10 \][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( 2x - 6 \)[/tex]:
[tex]\[ 5x + 10(2x - 6) = -10 \][/tex]
Simplify the equation:
[tex]\[ 5x + 20x - 60 = -10 \][/tex]
Combine like terms:
[tex]\[ 25x - 60 = -10 \][/tex]
Add 60 to both sides:
[tex]\[ 25x = 50 \][/tex]
Divide by 25:
[tex]\[ x = 2 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] at which the graphs of the equations intersect is [tex]\( \boxed{2} \)[/tex].
[tex]\[ \begin{cases} 2x - y = 6 \\ 5x + 10y = -10 \end{cases} \][/tex]
Let's go through the solution step by step:
1. Isolate one variable in one of the equations:
Start by isolating [tex]\( y \)[/tex] in the first equation:
[tex]\[ 2x - y = 6 \][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ -y = 6 - 2x \][/tex]
Then multiply by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 6 \][/tex]
2. Substitute the expression for [tex]\( y \)[/tex] into the other equation:
Now substitute [tex]\( y = 2x - 6 \)[/tex] into the second equation:
[tex]\[ 5x + 10y = -10 \][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( 2x - 6 \)[/tex]:
[tex]\[ 5x + 10(2x - 6) = -10 \][/tex]
Simplify the equation:
[tex]\[ 5x + 20x - 60 = -10 \][/tex]
Combine like terms:
[tex]\[ 25x - 60 = -10 \][/tex]
Add 60 to both sides:
[tex]\[ 25x = 50 \][/tex]
Divide by 25:
[tex]\[ x = 2 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] at which the graphs of the equations intersect is [tex]\( \boxed{2} \)[/tex].
