System’s Cheat
Sheet
Method:
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Step
1:
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Step
2:
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Step
3:
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| Graphing | Set each equation into slope intercept form (y = mx + b) | Plot the first point at (0,b) and use the slope to draw the line. Repeat for the second line | Find where they cross, this is the solution to the system (an ordered pair) | |
| Substitution |
Solve for one of the variables in one of the equations (get either x = or y = ) | Substitute this value into the OTHER equation in place of that variable and solve for the OTHER variable | Substitute this value back in for the first variable to get the ordered pair | |
| Add/Subtract |
Make sure both equations are in
standard form
(ax + by = c) |
Multiply each equation by the opposite leading coefficient so they are the same. If needed, change the signs so that one variable drops out. Solve for the remaining variable | Substitute this value back into
either equation to solve for the other variable.
This ordered pair is the solution to the system
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| Cramer’s
Rule |
Make sure both
equations are in standard form
(ax + by = e cx + dy = f) |
Identify the values for a, b, c, d, e, and f |
“Plug and Chug” |
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X = |
de – bf |
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ad - bc |
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Y = |
af - ce |
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ad – bc |
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| SIDE NOTES: |
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| A system of two linear equations will have either no solution, one solution, or all solutions. | Parallel lines have no intersection; therefore they have no solution since each solution is an ordered pair where the lines cross. Parallel lines have the same slope | Perpendicular lines have exactly one solution (as do all other lines that only cross once), but they have “special slopes” that are negative reciprocals of each other | Coinciding lines are lines that have all solutions in common (they overlap), therefore their solution is all real numbers, we also call this the identity Coinciding lines have the same slope | |
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