MAT140 WEEK 5 ALISHA
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American Public University *
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Course
105-001
Subject
Mathematics
Date
Feb 20, 2024
Type
docx
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4
Uploaded by PrivateTitaniumStingray35
1
Module 5 Assignment
Alisha Willis
MAT140
2
1. Definitions and Examples:
a. Solutions: refer to values of variables that satisfy all the given equations simultaneously. An example would be 2x−3y=52
x
−3
y
=5, the solution (x = 2, y = 1) makes the equation true.
b. System of Linear Equations: a set of two or more linear equations that involve the same variables. Example 3x+2y=103
x
+2
y
=10 and 2x−y=52
x
−
y
=5. The common solution to both equations forms the solution to the system.
2. Determining Solutions to a System:
x+y=8
x
+
y
=8 and 3x+2y=213
x
+2
y
=21:
a. For the ordered pair (2, 4): 2+4=6 and 3(2)+2(4)=6+8=14≠21.2+4=6 and 3(2)+2(4)=6+8=14
=
21. So, (2, 4) is not a solution.
b. For the ordered pair (5, 3): 5+3=8 and 3(5)+2(3)=15+6=21.5+3=8 and 3(5)+2(3)=15+6=21. So, (5, 3) is a solution.
2. Solve System of Linear Equations by Graphing:
x+y=4
x
+
y
=4 and x−y=2
x
−
y
=2 we can solve by graphing the two equations on the same coordinate plane. The point of intersection represents the solution, which is (3, 1).
3. Solve System of Linear Equations by Substitution:
Given y=3x+1
y
=3
x
+1 and 4y−8x=124
y
−8
x
=12, substitute the expression for y from the first equation into the second equation:
4(3x+1)−8x=12.4(3
x
+1)−8
x
=12.
x = 2. Substituting x back into the first equation gives y = 7. Therefore, the solution is (2, 7).
4. Addition Method for Solving a System of Equations:
3
For the addition method, we must add or subtract the equations to eliminate one variable, this makes it easier to solve for the other. Below are the steps:
1.
Write the equations with like terms aligned.
2.
Multiply one or both equations by a constant if needed to make the coefficients of a variable opposite.
3.
Add or subtract the equations to eliminate one variable.
4.
Solve the resulting equation for the remaining variable.
5.
Substitute the value back into one of the original equations to find the other variable.
5. Solve by Addition Method:
Given the system 4x+y=134
x
+
y
=13 and 2x−y=52
x
−
y
=5:
Multiply the second equation by 1 to align coefficients:
4x+y=134
x
+
y
=13 2x−y=52
x
−
y
=5
Adding the two equations eliminates y:
6x=18
⟹
x=3.6
x
=18
⟹
x
=3.
Substitute x = 3 into the second equation to find y:
2(3)−y=5
⟹
y=−1.2(3)−
y
=5
⟹
y
=−1.
The solution is (3, -1).
6. System of Three Linear Equations:
To create a system with the solution (2, 1, 5), one possible set is:
2x−y+z=32
x
−
y
+
z
=3 3x+2y−z=83
x
+2
y
−
z
=8 x+y+z=8
x
+
y
+
z
=8
By substituting x = 2, y = 1, and z = 5 into these equations, they are simultaneously satisfied.
7. Problem-Solving Steps:
a. Understand the Problem
b. Devise a Plan
c. Carry Out the Plan
d. Check Your Answer
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8. Triangle Angle Measures:
Let the smallest angle be x
x
. The largest angle is x+40
x
+40 and the remaining angle is x+20
x
+20. The sum of all angles in a triangle is 180°:
x+(x+40)+(x+20)=180.
x
+(
x
+40)+(
x
+20)=180.
Solving this equation gives x=40
x
=40, x+40=80
x
+40=80, and x+20=60
x
+20=60. Therefore, the angles are 40°, 80°, and 60°.