If $2010a + 2014b = 2018$ and $2012a + 2016b = 2020$, what is the value of $a - b$ ?
This problem involves a system of linear equations with two variables: $a$ and $b$. I want to solve for $a - b$, which is the difference of the variables. One way to do that is to eliminate one of the variables by adding or subtracting the equations. If I subtract the first equation from the second, I get $(2012a + 2016b) - (2010a + 2014b) = 2020 - 2018$. Simplifying the left side, I get $2a + 2b = 2$. Dividing both sides by 2, I get $a + b = 1$. Now I have a simpler equation relating $a$ and $b$. I can use this to express $a$ in terms of $b$ or vice versa. For example, if I subtract $b$ from both sides, I get $a = 1 - b$. Then, I can substitute this into the expression $a - b$ and get $a - b = (1 - b) - b$. Simplifying this, I get $a - b = 1 - 2b$. This is an expression for $a - b$ in terms of $b$ only. To find the value of $a - b$, I need to find the value of $b$. I can do this by plugging in $a = 1 - b$ into either of the original equations. For example, if I use the first equation, I get $2010(1 - b) + 2014b = 2018$. Expanding the parentheses, I get $2010 - 2010b + 2014b = 2018$. Combining like terms, I get $4b = 8$. Dividing by 4, I get $b = 2$. Now that I have $b$, I can plug it into the expression for $a - b$ and get $a - b = 1 - 2(2)$. Simplifying this, I get $a - b = -3$. 