Answered

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Which of these two systems appears easier to solve by substitution? Which of these two systems appears easier to solve by elimination?

1.
[tex]\[
\begin{cases}
-4x - 2y = 26 \\
y = -5x - 31
\end{cases}
\][/tex]

2.
[tex]\[
\begin{cases}
7x + 4y = 6 \\
-7x - 7y = -2
\end{cases}
\][/tex]

Sagot :

GinnyAnswer
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.
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