Systems of Equations
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- Primrose Brooks
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1 Sstems of Equations Sstem of equations two or more equations where ou want to find a sloution that makes all of them true simultaneousle (the same point makes them all true). Solution of a sstem of equations the point that makes all the equations true (satisfies all the equations, the point that all the lines go through). Three tpes of sstems of equations Sstems are classified as either inconsistent or consistent. Inconsistent means there is no solution to the sstem. Consistent means there is at least one solution. Consistent sstems come in two tpes, independent which means onl one solution, and dependent which means infinitel man solutions. No solution (same slopes, m = m ; different -intercepts, b b ) Lines are parallel the never intersect, there is no point that all the lines pass through, there is no solution. The equations are inconsistent because the do not have a solution. One solution (different slopes, m m ) Lines intersect, pass through one point. The equations are consistent because the have at least one solution. The equations are also independent because the onl have one point in common. Infinitel man solutions (same slope, m = m ; same -intercepts, b = b ) The lines are on top of each other, the cover each other, the are the same line. Ever point on one line is also on the other line so there are infinitel man points that satisf both equations although not ever point of the coordinate plane is a solution of the sstem of equations. The equations are consistent because the have at least one solution. The equations are also dependent because the have infinitel man points in common. The graphs coincide, the lines are coincident. Parallel lines same slopes, different -intercepts, lines never intersect. The smbol for parallel lines is. Perpendicular lines slopes are the negative reciprocals of each other. The intersect at a 0º angle (right angle). The smbol for perpendicular lines is. Nate's Notes page of
2 = -+ = The solution of this sstem of equations is the point where both lines intersect, (, ). These lines are perpendicular because the slope of the red line is - and the slope of the blue line is. The slope of one line is the negative reciprocal of the slope of the other line. Algebraicall this looks like m = -/m. This sstem of equations is consistent and independent. Nate's Notes page of
3 = (/) + = (/) These lines are parallel, the never intersect. The have the same slopes (slope of the red line is ½ and the slope of the blue line is ½ ) but different -intercepts (-intercept of red line is and the -intercept of the blue line is ). There is no solution to this sstem because there is no point where the two lines intersect. This sstem of equations is inconsistent. Nate's Notes page of
4 = + = You onl see one line on this graph because the lines are on top of each other. It does not look like the are the same line from the equations but when both equations are solved for (put in slope-intercept form) it is eas to see the are the same. There are infinitel man solutions to this sstem because ever point on one line is also on the other. Note that even though there are infinitel man points that make both equations true not ever point on the coordinate plane is a solution of the sstem of equations. For eample the point (0, 0) is not a solution to the sstem and neither is (, ). This sstem of equations is consistent, dependent, and coincident. Nate's Notes page of
5 = (/) + = (-/) This is a tpical sstem of equations, there is one point where the two lines intersect (-, ), this is the solution of this sstem of equations. This sstem of equations is consistent and independent. Nate's Notes page of
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