Question

elimination.

1. \\(\

$$\begin{cases}2x + 3y = 54\\\\4x + 7y = 85\\end{cases}$$

\\) 3. \\(\

$$\begin{cases}3x + 2y = 54\\\\4x + 5y = 85\\end{cases}$$

\\) 4. \\(\

$$\begin{cases}2x + 3y = 54\\\\-4x + 7y = 85\\end{cases}$$

\\)

1. use the elimination method to solve the system.

\\(\

$$\begin{cases}2x - y = 10\\\\10x + 4y = 5\\end{cases}$$

\\)

Explanation:

Problem 2:

Step1: Eliminate x by multiplying first equation by 2

Multiply \(2x + 3y = 54\) by 2: \(4x + 6y = 108\)

Step2: Subtract second equation from new first equation

\((4x + 6y) - (4x + 7y) = 108 - 85\)
\(4x + 6y - 4x - 7y = 23\)
\(-y = 23\)
\(y = -23\)

Step3: Substitute y into first equation to find x

\(2x + 3(-23) = 54\)
\(2x - 69 = 54\)
\(2x = 123\)
\(x = \frac{123}{2} = 61.5\)

Step1: Eliminate x by multiplying first equation by 4 and second by 3

First equation *4: \(12x + 8y = 216\)
Second equation *3: \(12x + 15y = 255\)

Step2: Subtract new first from new second equation

\((12x + 15y) - (12x + 8y) = 255 - 216\)
\(7y = 39\)
\(y = \frac{39}{7}\)

Step3: Substitute y into first equation to find x

\(3x + 2(\frac{39}{7}) = 54\)
\(3x + \frac{78}{7} = 54\)
\(3x = 54 - \frac{78}{7} = \frac{378 - 78}{7} = \frac{300}{7}\)
\(x = \frac{100}{7}\)

Step1: Add the two equations to eliminate x

\((2x + 3y) + (-4x + 7y) = 54 + 85\)
\(-2x + 10y = 139\) Wait, better to add: \(2x -4x +3y +7y = 139\) → \(-2x +10y = 139\). Alternatively, add the two equations as is: \(2x -4x +3y +7y = 54 +85\) → \(-2x +10y = 139\). Wait, actually, let's add:
First equation: \(2x +3y =54\)
Second: \(-4x +7y =85\)
Multiply first by 2: \(4x +6y =108\)
Now add to second: \(4x -4x +6y +7y = 108 +85\) → \(13y = 193\) → \(y = \frac{193}{13}\)

Step2: Substitute y into first equation to find x

\(2x + 3(\frac{193}{13}) = 54\)
\(2x = 54 - \frac{579}{13} = \frac{702 - 579}{13} = \frac{123}{13}\)
\(x = \frac{123}{26}\)

Answer:

\(x = 61.5\), \(y = -23\)

Problem 3: