SAT Algebra: The Balance Scale

A linear equation is a balance scale. Whatever you do to one pan you do to the other, and the goal is to get the variable sitting alone on its side. Algebra is the biggest math domain on the Digital SAT, and most of it comes back to that single picture.

The domain covers linear equations, systems of two equations, and inequalities — along with the word problems that hide those equations inside a story about tickets, gas mileage, or phone plans.

The method

• Translate before you solve. Name the unknown, then turn the English into symbols: “per” becomes multiplication, “3 more than” becomes + 3, “twice as many” becomes 2×. Most missed algebra points are translation errors, not arithmetic.
• Isolate the variable. Undo the operations in reverse, doing the same thing to both sides.
• Check the units. If the answer should be dollars and you produced dollars-per-hour, a step went sideways.

What turns them hard

Easy items ask you to solve for x. Hard ones ask what a piece of the equation means. Given a cost model C = 40 + 15h, the question wants what the 40 represents (the flat fee before any hours) or what the 15 does (the cost each additional hour adds). A second hard type asks whether a system has no solution or infinitely many — that turns on comparing the coefficients, not on grinding through the algebra. Same slope and different intercepts means parallel lines that never meet; same slope and same intercept means one line drawn twice.

Common questions

What does the Algebra domain include on the Digital SAT?

Linear equations in one and two variables, systems of linear equations, and linear inequalities — tested directly and through word problems that translate into them.

How do I tell if a system has no solution?

Compare the equations. Matching slopes with different intercepts give parallel lines and no solution; matching slopes with the same intercept give the same line and infinitely many solutions.