We have a system of two linear equations with two variables, $x$ and $y$: $20x + 30y = 210$ $80x + 50y = 490$ We want to find the values of $x$ and $y$ that satisfy both equations.

1. Problem Description

We have a system of two linear equations with two variables, $x$ and $y$:

$20x + 30y = 210$

$80x + 50y = 490$

We want to find the values of $x$ and $y$ that satisfy both equations.

2. Solution Steps

First, simplify the first equation by dividing by 10:

$20x + 30y = 210$

$2x + 3y = 21$ (Equation 1)

Also, simplify the second equation by dividing by 10:

$80x + 50y = 490$

$8x + 5y = 49$ (Equation 2)

Now, we want to eliminate one of the variables. Let's eliminate $x$. Multiply Equation 1 by -4:

$-4(2x + 3y) = -4(21)$

$-8x - 12y = -84$ (Equation 3)

Add Equation 3 to Equation 2:

$(8x + 5y) + (-8x - 12y) = 49 + (-84)$

$8x + 5y - 8x - 12y = 49 - 84$

$-7y = -35$

Divide both sides by -7:

$y = \frac{-35}{-7}$

$y = 5$

Now, substitute $y = 5$ into Equation 1:

$2x + 3(5) = 21$

$2x + 15 = 21$

$2x = 21 - 15$

$2x = 6$

$x = \frac{6}{2}$

$x = 3$

3. Final Answer

$x = 3$ and $y = 5$