Answer :
To determine which of the two systems is easier to solve by substitution and which is easier to solve by elimination, we will analyze the characteristics of each system.
### System 1:
[tex]\[ \begin{cases} -4x - 2y = 26 \\ y = -5x - 31 \end{cases} \][/tex]
### System 2:
[tex]\[ \begin{cases} 7x + 4y = 6 \\ -7x - 7y = -2 \end{cases} \][/tex]
#### Substitution Method:
In substitution method, we isolate one variable in one equation and substitute this expression into the other equation.
- System 1: In system 1, the second equation is already solved for [tex]\( y \)[/tex]:
[tex]\[ y = -5x - 31 \][/tex]
This makes system 1 easier to solve by substitution because we can directly substitute [tex]\( y \)[/tex] from the second equation into the first equation.
- System 2: In system 2, neither equation is directly solved for one variable. We would need to rearrange one of the equations to isolate [tex]\( x \)[/tex] or [tex]\( y \)[/tex].
Thus, System 1 is easier to solve by substitution.
#### Elimination Method:
In the elimination method, we make the coefficients of one variable equal (or opposites) and then add or subtract the equations to eliminate that variable.
- System 1: The coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the first equation are [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex] respectively, and in the second equation, they are [tex]\(-5\)[/tex] and [tex]\(-31\)[/tex]. These coefficients do not suggest any immediate simple multiples.
- System 2: In system 2, the coefficients of [tex]\( x \)[/tex] are [tex]\( 7 \)[/tex] and [tex]\(-7 \)[/tex]. These coefficients are already opposites which makes it straightforward to eliminate [tex]\( x \)[/tex] by adding both equations:
[tex]\[ (7x + 4y) + (-7x - 7y) = 6 + (-2) \][/tex]
[tex]\[ 0x - 3y = 4 \implies y = -\frac{4}{3} \][/tex]
This makes system 2 easier to solve by the elimination method.
Thus, System 2 is easier to solve by elimination.
### Conclusion:
- System 1 appears easier to solve by substitution.
- System 2 appears easier to solve by elimination.
### System 1:
[tex]\[ \begin{cases} -4x - 2y = 26 \\ y = -5x - 31 \end{cases} \][/tex]
### System 2:
[tex]\[ \begin{cases} 7x + 4y = 6 \\ -7x - 7y = -2 \end{cases} \][/tex]
#### Substitution Method:
In substitution method, we isolate one variable in one equation and substitute this expression into the other equation.
- System 1: In system 1, the second equation is already solved for [tex]\( y \)[/tex]:
[tex]\[ y = -5x - 31 \][/tex]
This makes system 1 easier to solve by substitution because we can directly substitute [tex]\( y \)[/tex] from the second equation into the first equation.
- System 2: In system 2, neither equation is directly solved for one variable. We would need to rearrange one of the equations to isolate [tex]\( x \)[/tex] or [tex]\( y \)[/tex].
Thus, System 1 is easier to solve by substitution.
#### Elimination Method:
In the elimination method, we make the coefficients of one variable equal (or opposites) and then add or subtract the equations to eliminate that variable.
- System 1: The coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the first equation are [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex] respectively, and in the second equation, they are [tex]\(-5\)[/tex] and [tex]\(-31\)[/tex]. These coefficients do not suggest any immediate simple multiples.
- System 2: In system 2, the coefficients of [tex]\( x \)[/tex] are [tex]\( 7 \)[/tex] and [tex]\(-7 \)[/tex]. These coefficients are already opposites which makes it straightforward to eliminate [tex]\( x \)[/tex] by adding both equations:
[tex]\[ (7x + 4y) + (-7x - 7y) = 6 + (-2) \][/tex]
[tex]\[ 0x - 3y = 4 \implies y = -\frac{4}{3} \][/tex]
This makes system 2 easier to solve by the elimination method.
Thus, System 2 is easier to solve by elimination.
### Conclusion:
- System 1 appears easier to solve by substitution.
- System 2 appears easier to solve by elimination.
