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SOML MEET 4
NAME: ___________________
EVENT 1
TEAM: ___________________
SYSTEMS OF LINEAR EQUATIONS SCHOOL: ___________________
1.
[2 Points]
Eight times a certain number added to five times a second number is 184.
The first number minus the second number is -3. Find the numbers.
ANS: ____________________
2.
[3 Points]
A landscaper has two kinds of solutions containing weed killer and water.
One is 5% weed killer and the other is 15% weed killer. The landscaper
needs 100 liters of a 12% solution and wants to make it by mixing the two
solutions. How much of each solution should be used?
ANS: ________ L of 5%
________ L of 15%
3.
[5 Points]
Solve the following system of equations:
3x + 2y - z = 0
5x - y - 8z = 9
x + 4y - 3z = -22
ANS: x = ______
y = ______
z = ______
SOML MEET 4
EVENT 1
SYSTEMS OF LINEAR EQUATIONS
1.
[2 Points]
NAME: _Key_________
Eight times a certain number added to five times a second number is 184.
The first number minus the second number is -3. Find the numbers.
Solution:
Let x = the first number and y = the second number
8x + 5y = 184
x - y = -3 ← multiply through by 5
8x + 5y = 184
5x - 5y = -15 Now add these two equations together:
13x
= 169
x = 169/13 = 13
Now substitute x = 13 into:
x - y = -3
13 - y = -3
13 + 3 = y
16 = y
The first number is 13.
The second number is 16.
ANS: ___13, 16_______
2.
[3 Points]
A landscaper has two kinds of solutions containing weed killer and water.
One is 5% weed killer and the other is 15% weed killer. The landscaper
needs 100 liters of a 12% solution and wants to make it by mixing the two
solutions. How much of each solution should be used?
Solution:
Let:
x = liters of the 5% weed killer solution
y = liters of the 15% weed killer solution
x + y = 100 The two solutions combined will total to 100 liters
0.05x + 0.15y = 0.12(100) = 12
The two solutions combined will have a total of 12 liters of weed killer.
Solve the first equation for y: y = 100 - x
Substitute y with (100-x) into the second equation:
0.05x + 0.15y = 12
0.05x + 0.15 (100 - x) = 12
0.05x + 15 - 0.15x = 12
- 0.10x = -3
x=3
Plug 30 in for x:
y = 100 - x = 100 - 30 = 70
We will need 30 liters of the 5% weed killer solutions and
70 liters of the 15% weed killer solution.
ANS: __30____ L of 5%
__70____ L of 15%
3.
[5 Points]
Solve the following system of equations:
3x + 2y - z = 0
5x - y - 8z = 9
x + 4y - 3z = -22
Solution:
Solve the first equation for z:
z = 3x + 2y
Use this expression to replace z in the other two equations of the system:
5x - y - 8z = 9
5x - y - 8(3x + 2y) = 9
5x - y - 24x - 16y = 9
x + 4y - 3z = -22
x + 4y - 3(3x + 2y) = -22
x + 4y - 9x - 6y = -22
-8x - 2y = -22 ← Divide by 2 to
simplify
-19x - 17y = 9
-4x - y = -11
Now we have a system of two equations:
-19x - 17y = 9
-4x - y = -11
Solve the second equation for y: y = -4x + 11
Substitute this into the other equation:
-19x - 17(-4x + 11) = 9
-19x + 68x - 187 = 9
49x = 196
x=4
Plug x = 4 into the equation previously solved for y:
y = -4x + 11 = -4(4) + 11 , so y = -5
Plug x = 4 and y = -5 into the equation previously solved for z:
z = 3x + 2y
z = 3(4) + 2(-5)
z=2
The solutions to the system of equations are:
ANS: x = ___4___
y = __-5___
z = ___2___