Question

which systems of equations are best solved using addition?
4x + 3y = 16
-4x - 5y = -16

4x + 3y = 16
y = 3x + 16

4x + 3y = -16
-4x - 5y = -16

4x + 3y = -16
3.1x + 5.1y = 16

3.1x - 5.1y = 16
3.1x - 5.1y = -16

Explanation:

Step1: Recall addition method criteria

The addition (elimination) method is optimal when one variable has coefficients that are opposites (sum to 0) or can easily be made opposites to eliminate a variable.

Step2: Analyze first system

System 1:
$4x + 3y = 16$
$-4x - 5y = -16$
The $x$-coefficients are $4$ and $-4$, which are opposites. Adding the equations will directly eliminate $x$.

Step3: Analyze second system

System 2:
$4x + 3y = 16$
$y = 3x + 16$
This system has one equation solved for $y$, so substitution is better.

Step4: Analyze third system

System 3:
$4x + 3y = -16$
$4x - 5y = -16$
The $x$-coefficients are identical, so subtraction (a variant of addition with multiplication by -1) can eliminate $x$, but direct addition does not.

Step5: Analyze fourth system

System 4:
$3.1x + 5.1y = 16$
$3.1x - 5.1y = -16$
The $y$-coefficients are $5.1$ and $-5.1$, which are opposites. Adding the equations will directly eliminate $y$.

Step6: Analyze fifth system

System 5:
$3.1x - 5.1y = 16$
$3.1x - 5.1y = -16$
Both equations have identical left sides but different right sides, so it is inconsistent, and addition would only show a contradiction, not solve for variables.

Answer:

1.

$$\begin{cases} 4x + 3y = 16 \\ -4x - 5y = -16 \end{cases}$$

2.

$$\begin{cases} 3.1x + 5.1y = 16 \\ 3.1x - 5.1y = -16 \end{cases}$$