Question

solve the following systems using any method.

1. $2x - y = 2$

$4x + 3y = 24$

1. $x = y + 1$

$2x - 2y = -5$

1. $y = 2x - 1$

$y = x + 1$

1. $6x + 21y = 132$

$6x - 4y = 32$

Explanation:

Problem 45

Step1: Solve for y from first equation

From \(2x - y = 2\), we get \(y = 2x - 2\).

Step2: Substitute y into second equation

Substitute \(y = 2x - 2\) into \(4x + 3y = 24\):
\(4x + 3(2x - 2) = 24\)
\(4x + 6x - 6 = 24\)
\(10x = 30\)
\(x = 3\).

Step3: Find y using x = 3

Substitute \(x = 3\) into \(y = 2x - 2\):
\(y = 2(3) - 2 = 4\).

Step1: Substitute x into second equation

Substitute \(x = y + 1\) into \(2x - 2y = -5\):
\(2(y + 1) - 2y = -5\)
\(2y + 2 - 2y = -5\)
\(2 = -5\) (Contradiction).

Step1: Set the two equations equal

Since \(y = 2x - 1\) and \(y = x + 1\), set \(2x - 1 = x + 1\).

Step2: Solve for x

\(2x - x = 1 + 1\)
\(x = 2\).

Step3: Find y using x = 2

Substitute \(x = 2\) into \(y = x + 1\):
\(y = 2 + 1 = 3\).

Answer:

\(x = 3\), \(y = 4\)

Problem 46