Unit 5 · Systems of Linear Equations
A solution to a system is an ordered pair that makes both equations true — the intersection of two lines, or the x where f(x) = g(x).
One intersection: one solution. Parallel distinct lines: no solution. Same line: infinitely many solutions.
Solve one equation for a variable, substitute into the other, then back-substitute. Best when a variable already has coefficient 1.
Multiply so coefficients of one variable are opposites, add the equations, solve, then find the other variable.
Two or more equations considered together.
Example: y = 2x + 1 and y = −x + 7 form a system.
An ordered pair that satisfies every equation in the system.
Example: (2, 5) solves y = 2x + 1 and y = −x + 7.
Try the prompt first. The steps stay folded until you ask to see them.
Solve y = 3x − 1 and 2x + y = 9.
Watch for
Stopping after finding x
A system solution is an ordered pair.
Fix: Always find both coordinates and check in both originals.
Submit each answer to check it. The key is not shown until you do.
Courtney spent $15.50 on four lattes and two donuts. The next day she spent $18.10 on three lattes and five donuts. If x is the cost of one latte and y is the cost of one donut, which system models this situation?
Dana went shopping for plants. She bought three roses and two daisies for $31.88. Later that day she bought two roses and one daisy for $18.92. If r is the cost of one rose and d is the cost of one daisy, which system models this situation?
The system y = x² − 3x − 6 and y = x − 1 is graphed. State the coordinates of the solution with the larger x-value as an ordered pair.
Graph first so the intersection is visible, then confirm algebraically.
Interpret a break-even system: cost C = 4x + 80 and revenue R = 12x.
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