SAT Math · Lesson 1 of 4
Heart of Algebra
Build and solve linear equations, systems, and inequalities, including relationships hidden in a word problem.
Create the equation before you calculate
Many algebra items are translation problems. Define the variable in a sentence: “p is the original price in dollars.” Then write the relationship. A 30% discount leaves 0.70 of the price, so a discounted price of 84 means 0.70p = 84, and p = 120.
Equivalent equations have the same solution, but equivalent expressions are equal for every allowed value. If 3x + 2 = 17, then 6x + 4 = 34 because the second expression is exactly twice the first. You do not need to solve for x first.
Systems and constraints
A system describes more than one condition. If a line and another line intersect, their x and y values satisfy both equations at that point. Graphically, the intersection is the solution. Algebraically, elimination or substitution finds the same point exactly.
An inequality describes a range. “No more than 5 hours” is h ≤ 5. On a graph, the solution of a linear inequality is a half-plane, and the boundary is solid when equality is allowed.
Worked example
A gym charges $20 per month plus $3 per visit. Another charges $8 per visit with no monthly fee. For how many visits is the first gym cheaper in one month?
- Let v be the number of visits. Costs are 20 + 3v and 8v.
- Set 20 + 3v < 8v, so 20 < 5v and v > 4.
- The first gym is cheaper for 5 or more visits. At 4 visits the costs are equal.
Why this works. The question asks for a comparison of two expressions, not for the value of either cost alone.
Check your understanding
- Write what the variable measures, including units.
- Use an equivalent-expression shortcut only when you can name the relationship.
- Check a system solution in both original equations.