### Instruction: A line $l$ is the graph of $y = ax + b$. Line $l_1$ is $l$ shifted 3 units to the left and two units downward and passes through the points $(5,8)$ and $(7,14)$. What is $a - b$?### Response: To shift $l$ three units to the left and two units downward, I need to subtract 3 from the x-coordinate and 2 from the y-coordinate of every point on $l$.
That means $l_1$ has the equation $y - 2 = a(x - 3) + b$.
I can use the points $(5,8)$ and $(7,14)$ to find $a$ and $b$.
Substituting the first point, I get $8 - 2 = a(5 - 3) + b$, which simplifies to $6 = 2a + b$.
Substituting the second point, I get $14 - 2 = a(7 - 3) + b$, which simplifies to $12 = 4a + b$.
I can solve this system of equations by subtracting the first equation from the second, which eliminates $b$ and gives me $6 = 2a$, so $a = 3$.
I can plug $a = 3$ back into either equation to find $b$.
For example, using the first equation, I get $6 = 2(3) + b$, so $b = 0$.
Therefore, $a - b = 3 - 0 = 3$.