Graphing and substitution
A system of equations asks for values that satisfy several equations at once. Graphing shows the answer as an intersection; substitution finds it exactly.
What a system is
A system of linear equations is a set of equations sharing the same variables, such as \(x + y = 10\) and \(x - y = 2\). A solution is an ordered pair that makes every equation true at once: here (6, 4), since 6 + 4 = 10 and 6 - 4 = 2.
Solving by graphing
Each linear equation in two variables graphs as a line, and a solution of the system is a point on both lines: their intersection. Graphing gives a clear picture and a quick estimate, but reading exact coordinates from a sketch is unreliable when the answer involves fractions.
Three possible outcomes
Two lines can meet at exactly one point, giving one solution; they can be parallel, never meeting, giving no solution (an inconsistent system); or they can be the same line, giving infinitely many solutions (a dependent system). Comparing slopes and intercepts tells you which case you have before you solve.
Substitution
Substitution solves one equation for one variable and substitutes that expression into the other. For \(y = 2x - 1\) and \(3x + y = 14\): substitute to get \(3x + (2x - 1) = 14\), so \(5x = 15\) and \(x = 3\). Then \(y = 2(3) - 1 = 5\). The solution is (3, 5).
Choosing what to isolate
Substitution works best when a variable already has coefficient 1 or -1. In \(x - 4y = 2\) and \(2x + 3y = 15\), isolate x in the first: \(x = 4y + 2\). Substituting gives \(2(4y + 2) + 3y = 15\), so \(11y = 11\), \(y = 1\), and \(x = 6\).
Spotting special cases
If the variable disappears, interpret the result. Substituting \(y = 3x + 1\) into \(6x - 2y = 4\) gives \(6x - 6x - 2 = 4\), so \(-2 = 4\): false, so no solution, and the lines are parallel. A true statement such as \(0 = 0\) means the equations describe the same line.
Checking
Always substitute the solution into both original equations. A pair that satisfies only one equation is a sign of an arithmetic slip, often a dropped negative sign when distributing.
Worked example
Cinema tickets cost $12 for adults and $8 for children. A group buys 9 tickets for $92. Let a be adults and c children: \(a + c = 9\) and \(12a + 8c = 92\). Substitute \(c = 9 - a\): \(12a + 72 - 8a = 92\), so \(4a = 20\), giving 5 adults and 4 children. Check: \(60 + 32 = 92\).
Key takeaways
A solution of a system satisfies every equation at once and is the intersection of their graphs. Two lines meet once (one solution), never (parallel, no solution), or everywhere (same line, infinitely many solutions). Graphing gives a picture; substitution gives exact answers by solving one equation for a variable and substituting into the other. Isolate a variable with coefficient 1 or -1 when possible. If the variable cancels, a false statement means no solution and a true one means infinitely many. Always check in both equations.
Flashcard
Equations sharing variables
Check your understanding
Solve \(y = 2x - 1\) and \(3x + y = 14\).