Topic 5.1 (Solving Linear Systems)
Systems of two linear equations.
1. Lines that are parallel
2. Lines that coincide
3 Lines that intersect
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Parallel Lines
x + y = 3 These two lines have the same slope, but different y-intercepts.
x + y = 1 They are parallel (same slope) and are independent and
inconsistent. They have no solutions. (ordered pairs that satisfy
both equations.) (Ø)
Graph both equations: The first one passes through the points (3, 0) & (0, 3).
The second one passes through the points (1, 0) & (0, 1)
Notice after connecting the points of each equation the two equations are
parallel and never touch each other.
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Lines that Coincide
x + y = 1 These two lines have the same slope and the same y-intercept.
2x + 2y = 2 They are the same line (the second is twice the first) and are
dependent and consistent. Answer: infinite number of ordered
pairs. Any ordered pair that works for the first equation, will also
work for the second equation.
Graph both equations: The first equation passes through the points: (0, 1) & (1, 0).
The second equation passes through the same points.
The first equation will have a solid line and the second
will have a broken line, which means the graph of the
second equation is on top of the first equation.
(in a two dimensional plane)
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Lines that intersect
x + y = 5 These two equations have different slopes, so they will intersect at
x - y = 1 one point (ordered pair). They have one solution and are
dependent and inconsistent.
Graph both equations: The first one passes through points: (0, 5), (5, 0).
The second one passes through points: (0, -1), (1, 0).
Notice that the two equations intersect at: (3, 2).
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Independent means something is different (slope or y-interceptd)
Parallel lines (same slope, different y-intercept)
Intersecting lines (different slopes, y-intercept can be same or different)
Dependent means same slope and same y-intercept.
Coinciding lines (same slope and same y-intercept)
Inconsistent means no order pair will satisfy both equations.
Parallel lines (no ordered pair will satisfy both equations)
Consistent means one or more ordered pair will satisfy both equations.
Coinciding lines (any ordered air that satisfies one equation will satisfy the
other.
Intersecting lines (one order pair satisfies both equations)
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Sample problems
Solve by graphing both equations:
x + 2y = -2 & 3x + y = 9 Solve each equation for y.
2y = -x + -2 y = -3x + 9 Determine m and b.
y = (-1/2)x + -1
m = (-1/2), b = -1 m = -3, b = 9
Label the intersecting kpoing.
Graph both equations: The first one passes through the points: (0, -1) & (-2, 0).
The second one passes through the points (0, 9) & (3, 0).
After drawing the lines as accurately as possible, you will see the intersecting
point. (4, -3)
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Use the substitution method to solve this system.
4x + y = 14 Take the first equation and solve for y in terms of x.
3x + 5y = -15 Substitute that value of y into the second equation.
y = -4x + 14, 3x + 5(-4x + 14) = -15 Solve for x
3x + -20x + 70 = -15
-17x = -85
x = 5 Substitute the value of x
into either of the two equations (the first equation is earsier).
4(5) + y = 14, 20 + y = 14, y = -6 (5, -6) Ans.
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All answers must be put in ordered pairs.
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Use the elimination method to solve this system.
4x - y = -8 The LCM for the x terms is 12. The LCM for the y terms is 2.
3x + 2y = 5 Use the smallest LCM.
Multiply the first equation by 2 to eliminate the y terms.
2(4x - y = -8) 8x - 2y = -16 Add both equations together to
3x + 2y = 5 3x + 2y = 5 eliminate the y terms.
11x = -11
x = -1
Substitute x into either equation (first one is the easiest) and solve for y.
4(-1) - y = -8, -4 - y = -8, -y = -4 (-1, -4) Ans.
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Solve this system: 12x - 3y = 132 Normally it is quicker to use the
6x + 5y = 14 elimination method. The only
time to use the substitution method is when one variable has a numerical
coefficient of one. In this problem multiply the second equation by -2
and add to the first equation.
12x - 3y = 132
-12x - 10y = -28 Solve for y
-13y = 104
y = -8 Substitute into the second equation to solve for x.
6x = 5(8) = 14, 6x - 40 = 14, 6x = 54, x = 9 (9. -8) Ans.
It is a good idea to check all problems using the ordered pair answer.