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Answer :
To solve the system of equations using the comparison method, we have:
1. [tex]\( x + y = 5 \)[/tex]
2. [tex]\( 2x + y = 7 \)[/tex]
The comparison method, in this case, involves manipulating the equations to eliminate one of the variables and comparing them.
First, let's rewrite the first equation for comparison:
From equation 1, we have:
[tex]\[ y = 5 - x \][/tex]
Now substitute this expression for [tex]\( y \)[/tex] into equation 2:
[tex]\[ 2x + (5 - x) = 7 \][/tex]
Simplify the equation:
[tex]\[
2x + 5 - x = 7
\][/tex]
This simplifies to:
[tex]\[
x + 5 = 7
\][/tex]
Now, let's solve for [tex]\( x \)[/tex]:
[tex]\[
x + 5 = 7 \implies x = 7 - 5 \implies x = 2
\][/tex]
Now that we've found [tex]\( x = 2 \)[/tex], we can substitute it back into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = 5 - x = 5 - 2 = 3
\][/tex]
Thus, the solution to the system is [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex].
Now, let's check which of the given equations corresponds with any parts of this process:
- [tex]\( 5 \cdot x = 7 \cdot 2x \)[/tex] does not match any of our transformation steps.
- [tex]\(-x - 5 = 7 - 2x\)[/tex] does not match either.
- [tex]\(5 - x = 2x - 7\)[/tex] can be related by reorganizing: [tex]\( x + 5 = 7 \)[/tex], which simplifies their difference in order, but doesn't directly match any correct steps.
Given this understanding, the answer to the original question is not present in the choices provided, as none of the manipulated forms directly lead to or align with those presented. However, during the logical steps following elimination of [tex]\( y \)[/tex], we found [tex]\( x +5=7 \)[/tex], which was a part of our process.
1. [tex]\( x + y = 5 \)[/tex]
2. [tex]\( 2x + y = 7 \)[/tex]
The comparison method, in this case, involves manipulating the equations to eliminate one of the variables and comparing them.
First, let's rewrite the first equation for comparison:
From equation 1, we have:
[tex]\[ y = 5 - x \][/tex]
Now substitute this expression for [tex]\( y \)[/tex] into equation 2:
[tex]\[ 2x + (5 - x) = 7 \][/tex]
Simplify the equation:
[tex]\[
2x + 5 - x = 7
\][/tex]
This simplifies to:
[tex]\[
x + 5 = 7
\][/tex]
Now, let's solve for [tex]\( x \)[/tex]:
[tex]\[
x + 5 = 7 \implies x = 7 - 5 \implies x = 2
\][/tex]
Now that we've found [tex]\( x = 2 \)[/tex], we can substitute it back into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = 5 - x = 5 - 2 = 3
\][/tex]
Thus, the solution to the system is [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex].
Now, let's check which of the given equations corresponds with any parts of this process:
- [tex]\( 5 \cdot x = 7 \cdot 2x \)[/tex] does not match any of our transformation steps.
- [tex]\(-x - 5 = 7 - 2x\)[/tex] does not match either.
- [tex]\(5 - x = 2x - 7\)[/tex] can be related by reorganizing: [tex]\( x + 5 = 7 \)[/tex], which simplifies their difference in order, but doesn't directly match any correct steps.
Given this understanding, the answer to the original question is not present in the choices provided, as none of the manipulated forms directly lead to or align with those presented. However, during the logical steps following elimination of [tex]\( y \)[/tex], we found [tex]\( x +5=7 \)[/tex], which was a part of our process.
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