Systems of linear equations word problems

How the SAT tests this, with worked examples — plus the 1 skills it covers. Free to read, no account needed.

These give a situation with two unknowns and enough information for two equations. The modelling is the work; solving is routine once the equations are written.
Start by naming the unknowns explicitly — let cc be the number of children and aa the number of adults — because most errors come from losing track of what a letter stands for.
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Then look for the two different kinds of information a question supplies. One equation usually counts items, the other totals values such as cost or weight.
So for 1212 tickets costing $108\$108 in total, with child tickets at $6\$6 and adult at $12\$12, the count equation is c+a=12c + a = 12 and the value equation is 6c+12a=1086c + 12a = 108.
Solve by substitution or elimination, whichever needs less rearranging, then answer the question asked. Having found both values, check whether the question wanted one of them, their difference, or something else entirely.

Worked examples

1212 tickets cost $108\$108. Child tickets are $6\$6 and adult tickets $12\$12. How many were children?
Write a count equation and a value equation
c+a=126c+12a=108{c + a = 12 \qquad 6c + 12a = 108}
Substituting a=12−ca = 12 - c gives 6c+144−12c=1086c + 144 - 12c = 108, so −6c=−36-6c = -36 and c=8c = 8.
In the same problem, how many more children than adults were there?
With c=8c = 8 and a=4a = 4, the difference is 44. The question asked for a difference, not a count — and 88 will be among the choices for anyone who stopped early.

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