Given that $x+y = 7$ and $3x+y = 45,$ evaluate $x^2-y^2.$
To solve for $x^2-y^2,$ I need to find the values of $x$ and $y$ first. I can use the given equations to eliminate one of the variables and solve for the other. For example, if I subtract the first equation from the second, I get $2x = 38,$ so $x = 19.$ Then, I can plug in $x = 19$ into either equation to find $y.$ Let me use the first one: $19+y = 7,$ so $y = -12.$ Now that I have $x = 19$ and $y = -12,$ I can plug them into the expression $x^2-y^2.$ I get $19^2 - (-12)^2 = 361 - 144 = 217.$ 