b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Students should know how to algebraically and graphically solve systems of linear equations in two variables. Tell them not to worry—it sounds way more complicated than it actually is.

We can't really do much with an equation in two variables because even if we solve for one, we just end up with an answer in terms of the other. Instead, we can turn two two-variable equations into a one-variable equation that we can actually solve until the very end.

The easiest way to solve a system is to solve either equation for either variable—get one x or y all by itself in one equation. Then, the value of that variable is substituted into the other equation to make an equation with only one variable that can easily be solved. Then, once the answer is found, it can be plugged into one of the equations to find the value of the remaining variable.

Students can also add or subtract equations together in order to cancel out one of the variables. We recommend holding off on this until students feel comfortable with substitution. That way, they'll have a better grasp of why we're allowed to do this, and a method to check their answers. Two birds with one stone, if you ask us.

Graphically, if students plot the two lines on the coordinate plane, they should be aware that the intersection point represents the solution to both linear equations. Students should not only understand why this is the case, but also verify that this is the case by plugging in the coordinates of the intersection point into each equation.

Students should also be able to estimate solutions just by looking at the equations. Tell students to look for similarities between the two equations (like how 3x + 2y = 5 and 3x + 2y = 6 have the same left side of the equation exactly). They should also know that two unique equations with the same slope will not share any solutions because the lines are parallel and will never intersect. (This is particularly useful when the equations are in standard form.)

Students should tighten their algebraic tool belts and make sure they're comfortable with 2 equations in 2 variables on 2 playing fields. They'll probably need 2 hands free (and hopefully it's not 2 much for them).