Question

1. solve the system using the substitution method.\\(- 5x + 3y = 51\\)\\(- 10x + y = - 8\\)

Explanation:

Step1: Solve the second equation for y

From \(-10x + y = -8\), we can express \(y\) in terms of \(x\) by adding \(10x\) to both sides.
\(y = 10x - 8\)

Step2: Substitute y into the first equation

Substitute \(y = 10x - 8\) into \(-5x + 3y = 51\).
\(-5x + 3(10x - 8) = 51\)

Step3: Simplify and solve for x

First, distribute the 3: \(-5x + 30x - 24 = 51\)
Combine like terms: \(25x - 24 = 51\)
Add 24 to both sides: \(25x = 51 + 24 = 75\)
Divide both sides by 25: \(x = \frac{75}{25} = 3\)

Step4: Substitute x back into y's equation

Now that we have \(x = 3\), substitute it into \(y = 10x - 8\).
\(y = 10(3) - 8 = 30 - 8 = 22\)

Answer:

The solution to the system is \(x = 3\) and \(y = 22\), or the ordered pair \((3, 22)\).