Key takeaway: A system of equations is a pair of equations in the same unknowns, and its solution is the one set of values that makes both statements true together.
Grade 88.EE.C.8Expressions and Equations
Definition of System of Equations
A system of equations puts two conditions on the same unknowns. Think of a ticket booth: you know how many tickets were sold and how much money came in, and you want to know how many of each kind. Each fact gives one equation, and only the right pair of numbers satisfies both.
For two lines on a graph there are three possibilities. The lines can cross at one point, giving exactly one solution. They can be parallel, giving none. They can be the same line, giving infinitely many. Most word problems at this level end with a single answer, found by graphing, by substitution or by elimination. Whichever route you take, the system of equations is solved only when the pair works in both statements.
Formula and notation
Two linear equations are usually stacked, one above the other:
\[ax + by = c\]\[dx + ey = f\]
A solution is an ordered pair \((x, y)\) that fits both lines at once.
Key facts
- The solution is the point where the graphs intersect.
- Parallel lines mean the system has no solution.
- Identical lines mean every point on the line is a solution.
- Always check the answer in both original equations.
System of Equations example
At a Friday matinee, grown-ups pay \$12 and kids pay \$8. The box office sold 7 seats for \$72 altogether. Use a system of equations, \(a + c = 7\) and \(12a + 8c = 72\), to find how many of each seat were sold.
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From the first equation, \(c = 7 – a\).
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Substitute: \(12a + 8(7 – a) = 72\), so \(4a + 56 = 72\) and \(a = 4\).
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Then \(c = 7 – 4 = 3\).
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Check: \(4 + 3 = 7\) and \(12(4) + 8(3) = 48 + 24 = 72\).
The box office sold 4 grown-up seats and 3 kid seats, the point \((4, 3)\).
Common mistake
Watch out
After finding \(a = 4\), Dario stops and says the solution of the system is 4.
Why it is wrong
There are two unknowns, so the solution is an ordered pair with both values. Substituting back gives \(c = 3\), and the solution is \((4, 3)\). Half an answer cannot be checked in both equations.
Related terms and lessons
- Grade 8 · Systems of Linear EquationsLessonWorksheetQuiz
- Algebra 1 · Systems of Linear EquationsLessonWorksheetQuiz
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