To solve the system of equations using the substitution method, we need to follow a step-by-step approach. Let's start with the given equations:
1. [tex]\( x + y = 12 \)[/tex]
2. [tex]\( -x = -y - 10 \)[/tex]
First, let's simplify the second equation for use in substitution. We start with:
[tex]\[ -x = -y - 10 \][/tex]
We can rewrite this equation by multiplying both sides by -1 to get:
[tex]\[ x = y + 10 \][/tex]
Now, we can substitute this expression for [tex]\( x \)[/tex] into the first equation:
[tex]\[ x + y = 12 \][/tex]
Substituting [tex]\( x = y + 10 \)[/tex] into the equation:
[tex]\[ (y + 10) + y = 12 \][/tex]
Combining like terms:
[tex]\[ 2y + 10 = 12 \][/tex]
Next, solve for [tex]\( y \)[/tex]:
[tex]\[ 2y + 10 - 10 = 12 - 10 \][/tex]
[tex]\[ 2y = 2 \][/tex]
[tex]\[ y = \frac{2}{2} \][/tex]
[tex]\[ y = 1 \][/tex]
Now that we have the value of [tex]\( y \)[/tex], we can substitute it back into the equation [tex]\( x = y + 10 \)[/tex] to find [tex]\( x \)[/tex]:
[tex]\[ x = 1 + 10 \][/tex]
[tex]\[ x = 11 \][/tex]
Therefore, the solution to the system of equations is the ordered pair:
[tex]\[ (x, y) = (11, 1) \][/tex]
Let's check whether this solution matches any of the given options:
A. [tex]\((11,1)\)[/tex]
B. [tex]\((9,3)\)[/tex]
C. [tex]\((8,4)\)[/tex]
D. [tex]\((10,2)\)[/tex]
The correct ordered pair is:
A. [tex]\((11, 1)\)[/tex]
