Algebra 1 — linear equations, systems and inequalities

Thirteen questions covering all five of the skill points College Board lists under Algebra: linear equations in one variable, linear equations in two variables, linear functions, systems of two linear equations, and linear inequalities.

The mix is deliberate, because the domain is not one kind of question.

Pure algebra. Solve, rearrange, find a slope from an equation in standard form. These are the quickest questions on the Math section and the ones to get out of the way fast, because the time they save is what you spend on the hard ones later in the module.

Structure, not arithmetic. Two questions here have unknown coefficients and ask when a system has no solution and when it has infinitely many. Both are much faster answered as facts about lines than as algebra: no solution means the same slope and a different intercept — parallel lines; infinitely many means one equation is a multiple of the other — the same line written twice. A student who solves rather than recognises will get there, eventually, having spent two minutes.

Interpretation. Two questions ask what a coefficient or a constant represents in a real situation and require no calculation at all. In $y=15x+100$ the 100 is what you have before anything happens and the 15 is what happens per unit. Free marks, and routinely lost by students who start substituting.

Word problems where the difficulty is the setup. The printing question here is the archetype: the arithmetic is one subtraction and one division, and three of the four options are the places you can go wrong — using the total instead of the total minus the setup fee, adding the fee rather than subtracting it, and giving the intermediate value in dollars when the question asked for a number of pages. That last one is the most common wrong answer on this domain in general, and the explanations name it every time it appears.

One note on how these are written. Every price is spelled out in words rather than written with a dollar sign, because a dollar sign is also the delimiter for mathematical notation and the two collide. On the real test you will see currency symbols; here you will see "40 dollars", and nothing about the mathematics changes.

The twelve flashcards are the facts this domain runs on: the two forms of a linear equation, the slope formula, what "at most" and "at least" translate to, and the two conditions on systems written as sentences you can recite.

• Solve a linear equation in one variable and recognise when it has no solution or infinitely many
• Find the slope of a line given in standard form, and move between forms
• Decide when a system of two linear equations has no solution, one solution, or infinitely many — from the coefficients, without solving
• Interpret the coefficient and the constant of a linear model as quantities in a real situation
• Translate a word problem into an equation by defining the variable before writing anything
• Check what the question asked for before choosing — the intermediate value is usually among the options
• Translate "at most" and "at least" into the correct inequality sign

In the system of equations 3x + 6y = 12 and ax + 4y = 9, a is a constant. For what value of a does the system have no solution? — A system of two linear equations has no solution when the lines are parallel but not identical: same slope, different intercept. Matching the ratio of coefficients gives a = 2, and at a = 2 the second equation reduces to x + 2y = 4.5 against the first's x + 2y = 4 — parallel and distinct.