Question

solving a system of two equations in two unknowns by elimination can be done by adding or subtracting one equation from the other.
elimination by adding
solve the system: \\(x + 4y = 8\\)\\(3x - 4y = 8\\)
solution
notice that the terms \+4y\ and \-4y\ are opposites. this means that the two equations can be added without changing the signs.\\(\

$$\begin{align}x + 4y &= 8\\\\3x - 4y &= 8\\\\\\hline 4x + 0 &= 16\\end{align}$$

\\)\\(4x = 16\\), or \\(x = 4\\)
substitute \\(x = 4\\) in either of the equations to find \\(y\\). \\(x + 4y = 8\longrightarrow 4 + 4y = 8\\)\\(4y = 4\\), or \\(y = 1\\)
the solution of this system is \\((4, 1)\\).
elimination by subtracting
solve the system: \\(2x - 5y = 15\\)\\(2x + 3y = -9\\)
solution
notice that the terms \2x\ are common to both equations. however, to eliminate them, it is necessary to subtract one equation from the other. this means that the signs of one equation will change. here, the top equation stays the same. the signs of the bottom equation change.\\(\

$$\begin{align}2x - 5y &= 15\\\\(-)2x (-)3y &= (+)9\\\\\\hline 0 - 8y &= 24\\end{align}$$

\\) or \\(y = -3\\)
substitute \\(y = -3\\) in either of the original equations to find \\(x\\):\\(2x - 5y = 15\longrightarrow 2x - 5(-3) = 15\\)\\(2x + 15 = 15\\), or \\(x = 0\\)
the solution of this system is \\((0, -3)\\).
solve the following systems by elimination. state whether addition or subtraction is used to eliminate one of the variables.

1. \\(\

$$\begin{cases}3x + 2y = 10\\\\3x - 2y = 14\\end{cases}$$

\\)
operation:
solution: (, )

1. \\(\

$$\begin{cases}x + y = 12\\\\2x + y = 6\\end{cases}$$

\\)
operation:
solution: (, )

Explanation:

Problem 1

Step1: Identify elimination operation

The system is \(

$$\begin{cases} 3x + 2y = 10 \\ 3x - 2y = 14 \end{cases}$$

\). The \( +2y \) and \( -2y \) are opposites, so we use addition to eliminate \( y \).

Step2: Add the two equations

\( (3x + 2y) + (3x - 2y) = 10 + 14 \)
\( 6x + 0 = 24 \)
\( 6x = 24 \)

Step3: Solve for \( x \)

\( x = \frac{24}{6} = 4 \)

Step4: Substitute \( x = 4 \) into first equation

\( 3(4) + 2y = 10 \)
\( 12 + 2y = 10 \)
\( 2y = 10 - 12 = -2 \)
\( y = \frac{-2}{2} = -1 \)

Step1: Identify elimination operation

The system is \(

$$\begin{cases} x + y = 12 \\ 2x + y = 6 \end{cases}$$

\). The \( +y \) terms are the same, so we use subtraction (subtract the first equation from the second) to eliminate \( y \).

Step2: Subtract the first equation from the second

\( (2x + y) - (x + y) = 6 - 12 \)
\( 2x + y - x - y = -6 \)
\( x + 0 = -6 \)
\( x = -6 \)

Step3: Substitute \( x = -6 \) into first equation

\( -6 + y = 12 \)
\( y = 12 + 6 = 18 \)

Answer:

Operation: Addition; Solution: \( (4, -1) \)

Problem 2