Liam’s football team scored a total of 43 separate times this season, with a mix of touchdowns and field goals. each touchdown is worth 7 points, and each field goal is worth 3 points. if the team scored a total of 301 points this season, how many touchdowns and field goals did they score?

This question is an example of what we call "system of equations". We can use two separate equations to help us solve. 

Let a = number of touchdowns
Let b = number of field goals

a + b = 43
7a + 3b = 301

We can solve this system using the elimination method. 

1. a + b - b = 43 + -b (add - b to both sides)
     a = -b + 43

2. substitute -b + 43 for a in 7a + 3b = 301
     7 (-b +43) + 3b = 301
     -4b + 301 = 301 (simplify both sides of the equation)
      -4b + 301 - 301 = 301 - 301 (add -301 to both sides)
        -4b = 0
          b  = 0 

Since b = 0, and b is the number of field goals, we know that Liam's team did not score any field goals. Which means the only points from his team came from touchdowns. So, 43 touchdowns were scored, and 0 field goals were scored, for a total of 301 points.