SAT Topics: Algebra

Officially, this domain is divided into five skills, but there's a great deal of overlap among them, and it can sometimes be difficult to tell which skill you are dealing with. It's all just linear algebra. With many of the problems in the easy category, you may be able to solve them in your head, and if not, you can often solve them in a few seconds using Desmos. Many of these questions involve interpreting the meaning in various parameters: slopes and coefficients correspond to rates of increase, constant numbers and y-intercepts correspond to initial starting values, and so on. Here are the College Board's five official sub-categories in this domain:

Linear Equations (one variable) — About half of the problems in this category have to do with "interpretation", i.e. translating from words to symbols or vice versa. Most of the rest involve solving simple equations. Usually you can solve these in a couple of seconds using Desmos. Sometimes they try to prevent you from using Desmos by asking which equation has the same solution?, but that's not much of an obstacle. Some of the problems ask how many solutions the equation has. All normal equations should have one solution, but if you are careless with symbols, it is possible to write down equations that have no solutions, like 7=5, or equations that have an infinite number of solutions, like 7=7. In these cases, do whatever simplification seems appropriate, and then decide if the two sides of the equation are always the same or never the same. Here are the easy, medium, and hard practice problems from the SAT Practice Problem Databank.

Linear Functions — Sometimes they give you a function and an input value, and you simply have to evaluate the function for the given value. But beware. Sometimes they give you the output value. They tell you that the function is equal to something, and you have to work the function backwards to find the corresponding input value ... but this is no different from solving equations. Problems involving graphs are common, so be able to interpret slopes and intercepts. Here are the easy, medium, and hard practice problems from the SAT Practice Problem Databank.

Linear Equations (two variables) — These are equations with both an x and a y, and many of these problems are indistinguishable from function problems. Modeling and interpretation problems in which you need to translate from words to symbols or vice versa are common. Equations in slope-intercept form are very common, and it is very helpful to be able to calculate slopes and intercepts, and to interpret the slope and intercept physically. (The slope or coefficient is always a rate of some kind, and the intercept or constant term is probably an initial value of some kind.) Here are the easy, medium, and hard practice problems from the SAT Practice Problem Databank.

Systems of Equations — There are two equations and two variables, and you either need to interpret what they stand for in words, or you need to solve the system. If you need to solve a system of equations, it is often (though not always) easy to solve with Desmos. Sometimes they ask about the number of solutions to the system. All normal linear systems should have one solution, but if you are sloppy with the symbols, it is possible to write down a system with no solutions (the lines are parallel when you graph them) or an infinite number of solutions (they are equivalent equations and yield the same line when you graph them). Here are the easy, medium, and hard practice problems from the SAT Practice Problem Databank.

Linear Inequalities — These are virtually the same as linear equations in one or two variables, but with an inequality symbol instead of an equals sign. You will usually either need to interpret something or solve something, and you can often solve inequalities simply using Desmos. In interpretive problems, the direction of the arrow gives you another quick check that you can try. You can probably rule out two wrong answers for having the arrow pointing in the wrong direction. If you have a system of inequalities and you need to find which point satisfies the pair of inequalities, you can do it quickly and easily in Desmos. (You can enter a coordinate pair into Desmos, and it will draw a dot at that point.) Here are the easy, medium, and hard practice problems from the SAT Practice Problem Databank.