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Solving Systems of Linear Equations Graphically


Sometimes more than one equation, each representing a different scenario, can be written using the same variables. Consider the following scenario.

Erin is contemplating purchasing a more efficient furnace for her house. She expects the new furnace will reduce her average monthly heating bill from \(\$190\) to \(\$120\). The furnace will cost \(\$3\thinspace 400\) to purchase and install. This information can be summarized with two equations.

  • The cost, \(c\), to continue running the old furnace for \(m\) months: \(c = 190m\)
  • The cost, \(c\), to install and run the new furnace for \(m\) months: \(c = 120m + 3 \thinspace 400\)

This is an example of a system of equations.

When a system of linear equations is graphed, the point of intersection of the lines corresponds to the solution to the system. For example, the solution to the two-furnace system of equations is \((48.57, 9\thinspace 228.57)\). After \(48.57\) months, the two furnaces will cost the same amount, \(\$9\thinspace 228.57\).



The solution to a system can be verified using substitution.

\(c = 190m\)

Left Side Right Side
\(\begin{array}{r}
 c \\
 9\thinspace 228.57  \end{array}\)
\(\begin{array}{l}
 190m \\
 190(48.57) \\
 9\thinspace 228.3 
 \end{array}\)
                LS \(\doteq\) RS


\(c = 120m + 3 \thinspace 400\)

Left Side Right Side
\(\begin{array}{r}
 c \\
 9\thinspace 228.57  \end{array}\) 		\(\begin{array}{l}
 120m + 3\thinspace 400\\
 120(48.57) + 3\thinspace 400 \\
 9\thinspace 228.4 
 \end{array}\)
                LS \(\doteq\) RS



The solution satisfies each equation. The solution is verified.

The solution \((48.57,9 \thinspace 228.5)\) contains rounded values, so the verifications show only approximately equal values.