We are given a system of two linear equations: $4p + 2q = 62$ $8p - q = 59$ We need to determine if the two proposed methods for solving the system are valid. Method 1: Multiply the first equation by 2, then subtract the second equation from the result. Method 2: Multiply the second equation by 2, then add the result to the first equation.

1. Problem Description

We are given a system of two linear equations:

$4p + 2q = 62$

$8p - q = 59$

We need to determine if the two proposed methods for solving the system are valid.

Method 1: Multiply the first equation by 2, then subtract the second equation from the result.

Method 2: Multiply the second equation by 2, then add the result to the first equation.

2. Solution Steps

Let's analyze each method.

Method 1:

First equation multiplied by 2:

$2(4p + 2q) = 2(62)$

$8p + 4q = 124$

Subtracting the second equation from the result:

$(8p + 4q) - (8p - q) = 124 - 59$

$8p + 4q - 8p + q = 65$

$5q = 65$

$q = 13$

Substituting $q = 13$ into the first equation:

$4p + 2(13) = 62$

$4p + 26 = 62$

$4p = 36$

$p = 9$

Method 2:

Second equation multiplied by 2:

$2(8p - q) = 2(59)$

$16p - 2q = 118$

Adding the result to the first equation:

$(16p - 2q) + (4p + 2q) = 118 + 62$

$16p - 2q + 4p + 2q = 180$

$20p = 180$

$p = 9$

Substituting $p = 9$ into the first equation:

$4(9) + 2q = 62$

$36 + 2q = 62$

$2q = 26$

$q = 13$

Both methods yield the same solution for $p$ and $q$.

The original system is:

$4p + 2q = 62$

$8p - q = 59$

Substituting $p = 9$ and $q = 13$ into both equations to verify the solution.

$4(9) + 2(13) = 36 + 26 = 62$

$8(9) - 13 = 72 - 13 = 59$

Thus, both methods are valid.

3. Final Answer

Both strategies work for solving the system. The solution is

$p = 9$

$q = 13$