Section 3-5
Systems of equations p. 124
When would a system have a single solution? an infinite number of solutions? No solutions?
Single solution – the lines cross in one place. If you solve the equations for y = mx + b, both the slope and y-intercept are different.
Infinite solutions – they are the same line. We also call this equivalent. If you solve both for y = mx + b, they will be exactly the same
No solutions – The lines are parallel, and don’t cross, so they share no solutions. If you solve for y=mx+b, then they have the same slope, but a different y –intercept.
Solving systems of equations:
Solve the system: 4x + 3y = 5 Decide whether you will use
2x - 5y = -17 substitution or elimination
Probably easier to use elimination; multiplying the second equation by -2 would produce a system in which the x values would eliminate.
4x + 3y = 5
-4x +10y = 34 Make sure to multiply both sides by -2
13y = 39 Add the equations, producing a linear combination
y = 3 Solve for y
4x + 3(3) = 5 Substitute to find x, either equation is fine
4x = -4
x = -1
(-1, 3) This is the point that both equations share. If you were to graph the equations, this is where they would cross.
Solutions to systems are points.
Compare the Linear-Combination (elimination) to substitution method.
Linear-Combination (or elimination) Substitution
x - 2y = 5 multiply the top by -4 x - 2y = 5 Solve the top for x
4x + 3y =9 4x + 3y =9
-4x + 8y = -20 x = 2y + 5
4x + 3y = 9 2nd eq. remains the same
11y = -11 Add the eq.'s 4(2y+5) +3y = 9 Sub. in 2y + 5 for x
y = -1 Solve for y 8y + 20 + 3y = 9 Solve for y
x - 2(-1) = 5 Sub. to find x 11y = -11
x +2 = 5 y = -1
x = 3 x + 2 = 5 Sub. to find x
x = 3
Either way: Solution (3, -1)
Practice.
1. 2x + y =7
3x - y = 8
If you add the equations 5x = 15 and x = 3
2 (3) + y = 7; y = 1; solution is (3,1)
2. 4x - y = 9
3x - 5y = 11
-20x + 5y= -45
3x – 5y = 11
-17x = -34
X = 2 4(2) – y = 9; y = -1 solution (2, -1)
3. a + b = 9
a - b = 13
Add the equations and 2a = 22; a = 11, substitute and b = -2
Solution is in alphabetical order (11, -2)
4. 3x + 2y = 4
2x - 5y = -29
15x + 10y = 20
4x – 10y = -58
19x = -38
x = -2 3 (-2) + 2y = 4
2y = 10; y = 5 Solution (-2, 5)
5. 3x - 2y = x + 8
y = x – 4
Substitute x -4 for y in the first equation; easiest since second equation is solved for y
3x – 2(x – 4) = x + 8
3x – 2x + 8 = x + 8
x + 8 = x + 8
0 = 0 infinite solutions, since it is a true statement
What occurs when the lines are parallel, meaning that there is no solution?
Solve the system x + 2y = 1
y = -12x + 4
Using substitution x + 2[(-1/2)x + 4] = 1
x - x + 8 = 1
8 = 1 Untrue statement = no solution = parallel lines
You could see that they are parallel if you re-arranged either the first equation into
slope-intercept or the second equation into standard.
x + 2y = 1
2y = -x + 1
y = -12x + 1/2 Same slope but different y-intercept, so lines are parallel
What occurs when there are infinite solutions, or they are the same line?
Solve the system -3x + 5y = -6 You may notice right away that they are
6x -10y =12 the same line; consider dividing the second line by -2.
If you were to solve by elimination, you would either multiply the first equation by
2, or divide the second equation by 2. In the first case...
-6x +10y = -12
6x -10y = 12
0=0 This is a true statement. This occurs if there are infinite solutions
because the two equations represent the same line
Practice...
1. 2x + 3y = 13
x - y = 9
elimination: 2x + 3y = 13
3x – 3y = 27
5x = 50
X = 10 10 – y = 9; y = 1 solution (10.9)
2. 2x - y = 7
4x - 2y = 9
Elimination -4x + 2y = -14
4x – 2y = 9
0 = -5 not true, so lines parallel, and no solution
3. 3x - 4y = 14
4x + 10y = 11
Elimination 30x – 40y = 140
16x + 40y = 44
46x = 184
X = 4 3(4) – 4y = 14;
-4y = 2 y = -1/2 Solution (4, -1/2)
4. 6x = 4y + 5 Solve both for y and you see that they are parallel and not the same line.
6y = 9x - 5
y = 23x+ 56 and y = 23x- 56
No solution
5. x + y = 3x -1
x - y = 1 - x
put in standard form first (you could also solve one or both equations and use substitution)
-2x + y = -1
2x – y = 1
0 = 0 Infinite solutions – same line when 0=0
6. x + y = 4(y + 2)
x - y = 2(y + 4)
put both in standard form
x - 3y = 8
x – 3y = 8 Same line, infinite solutions
Section 3.6
Develop systems to solve these problems, even if you can do them in one variable.
1. To use a certain computer data base, the charge is $30/hr during the day and $10.50/hr at night. If a research company paid $411 for 28 hours of use, find the number of hours charged at the daytime rate, and at the nighttime rate. Hint: Use one equation to represent hrs, and a second to represent money.
Check your answer on p. 131
2. To measure the speed of the jet stream (a high-speed, high-altitude west-to-east wind), a weather-service plane flew 1800 km with the jet stream as a tail wind and then back again. The east bound flight took 2 hrs, and the westbound flight took 3 hrs 20 minutes. Find the speed of the jet stream, and the speed of the plane.
Thoughts....
a. You will use the distance formula
b. You will develop one equation for east bound and a second for ?westbound
c. Realize that a tail wind will speed up a plane, and a head wind will ?slow it down
check your answer on p. 132
Practice Problems
1. If 8 markers and 7 pencils cost $3.37 and 5 markers and 11 pencils cost $3.10, how much does each markers and each pencil cost?
8m + 7p = 3.37 multiply by 11 88m + 77p = 37.07
5m + 11p = 3.10 multiply by -7 -35m - 77p = -21.70
53m = 15.37
M = 0.29 markers
8(.29) + 7p = 3.37
7p = 1.05
P = 0.15 pencils
2. Lauren's marble jar contains plain marbles and colored marbles. If there are 32 more plain marbles than colored marbles, and there are 180 marbles in the jar, how many colored marbles does Loren have?
P = 32 + c equation representing that there are 32 more plain
P + c = 180 equation representing that the sum is 180
Use substitution
32 + c + c = 180
2c = 148
C = 74 colored marbles
Find P: P = 32 + 74 = 106 plain marbles
3. Ann can paddle her canoe 15 miles downstream in 2 hours. The return trip upstream takes three times as long. What is the rate of the current?
Upstream, the combined rate is r – c (the rate of the paddler minus the current)
Downstream, the combine rate is r + c (the rate of the paddler increased by the current)
Rate x time = distance
2(r + c) = 15 6(r – c) = 15 At this point, dividing the distance by time gives nice equations
r + c = 7.5 r – c = 2.5
Add the equations together and 2r = 10 and r = 5 mph, the rate of the paddler in still water
If r + c = 7.5 and r = 5, then c = 2.5 mph (rate of current)
4. A grocer mixed nuts worth $4 a kilogram with raisins worth $3.25 a kilogram to make 15 kilograms of mixture worth $3.60 per kilogram. How many kilograms of nuts were used?
Nice opportunity for a graphic organizer known as a chart!
Amount x / Cost = / Total costNuts / 15-x / 4 / 4(15-x)
Raisins / X / 3.25 / 3.25x
Mix / 15 / 3.60 / 54
What was given in the problem is in read. The blue I figured out by making one of the amounts x, the other must be the total minus x. 15 – x and x could have switched places.
The easiest way to do this is without a system at this point. In a sense, we have already done the substitution within the chart. See below in red for the systems method.
4(15-x) + 3.25x = 54
60 – 4x + 3.25x = 54
-.75x = - 6
X = 8 8 kg of raisins and 15-8, or 7 kg of nuts
Systems method
Nuts = n and Raisins = r
n + r = 15 this represents amount
4n + 3.25r = 54 this represents cost
Multiply the first equation by -4
-4n – 4r = -60
4n + 3.25r = 54
-.75r = -6
R = 6kg and N = 7 kg