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High School Mathematics
1.20 Linear Equations - Two Variables

Linear equation with one variable: has one solution.
Example:
x + 4 = 12
x + 2x + 6 = 9 simplifies to 3x + 6 = 9

Linear equation with two variables: A linear equation in two variables has infinitely many solutions.
Example:
x + y = 12
For x = 1, y = 11,
x = 2, y = 10,
x = 3, y = 9,
x = 4, y = 8..... and so on

Linear equation with three variables
Example:
x + y + z = 12

Note: You will need one equation to solve for one variable, two equations to solve two variable and three equations to solve three variable and so on......


Method 1:
Elimination Method:
Solving Linear Equations with two variables.

Solve for the values of a and b for the following equations:
Example 1:
a + b = 12 -----equation 1
a - b = 4 ------- equation 2. Adding both equations we get

a + b = 12
a - b = 4
------------------The terms +b and -b cancels, and we get
2a = 16
a = 16/8 = 2
a = 2
substituting the value of a in equation 1 we have
a + b = 12
2 + b = 12
b = 12 - 2
b = 10
The solution of the two equations is a = 2 and b = 10

Example 2:
c + 5b = 16 -----equation 1
c + 2b = 7 ------- equation 2 . Multiplying the equation 2 with (-1) we get

c + 5b = 16 -----equation 1
-1(c + 2b = 7) => -c - 2b = -7-------equation 2

c + 5b = 16
-c - 2b = -7
---------------------The terms c and -c cancel each other and we get
3b = 9
b = 9/3 = 3
substituting b in the equation c + 5b = 16 we get
c + 5(3) = 16
c + 15 = 16 - 3
c = 16 - 15
c = 1

Example 3:
2a + 4b = 12 -----equation 1
a - b = 3 ------- equation 2
First pick a variable either a or b. Lets pick b.
Second find the common multiple of b so we can eliminate that variable and find the value of the other variable.
so multiply both sides of the equation 2 by 4 we have
2a + 4b = 12 -----equation 1
4(a - b = 3) => 4a - 4b = 12------- equation 2

Now adding equation 1 and 2 we have
2a + 4b = 12
4a - 4b = 12
----------------------The terms +4b in equation 1 and -4b in equation 2 cancels, and we get
6a = 24
a = 24/6 =4
substituting value of a in equation a - b = 3 we get
4 - b = 3
4- 3 = b
b = 1
Therefore a = 4 and b = 1

Verification:
To verify substitute the values of a and b in the equations above
2a + 4b = 12
2a + 4b = 2(4) + 4(1) = 8 + 4 = 12

NOTE: In the above question if you pick b
2a + 4b = 12 -----equation 1
a - b = 3 ------- equation 2
multiplying equation 2 by 2 we have 2(a - b = 3) ------- equation 2
2a - 2b = 6 multiplying equation by -1 we gave -1(2a - 2b = 6) we get -2a + 2b = -6
Solving both equations we have
2a + 4b = 12 -----equation 1
-2a + 2b = -6-----equation 2
-------------------------
6b = 6
b = 1

Example 4:
2a + b = 4 -----equation 1
3a + 2b = 3 ------- equation 2. Multiplying the equation 1 with 3 and equation 2 with (-2) we get

3(2a + b = 4) -----equation 1
-2(3a + 2b = 3) ------- equation 2

6a + 3b = 12 -----equation 1
-6a - 4b = -6) ------- equation 2
-----------------------The terms 6a and -6a cancel each other and we get
-b = 6
b = -6
substituting b in the equation 2a + b = 4 we get
2a + (-6) = 4
2a - 6 = 4
2a = 4 + 6
2a = 10
a = 10/2 = 5

Method 2:
Substitution Method:
Solve the equations:
3x + 5y = 26
y = 2x
substitute the second equation in the first
3x + 5(2x) = 26
3x + 10x = 26
13x = 26
x = 2
y = 2x = 2x2 = 4

Solve the equations:
x + y = 26 ------- equation 1
x - y = 4 -------equation 2

x - y = 4 -------equation 2 can be written as x = y + 4.
Substituting x in the equation 1 we have:
x + y = 26 ------- equation 1
y + 4 + y = 26
2y + 4 = 26
2y + 4 - 4 = 26 - 4
2y = 22
y = 11
x = y + 4 = 11 + 4 = 15

Directions: Solve for the variables using either Elimination or Substitution Method.

Name: ___________________

Date:___________________

High School Mathematics
1.20 Linear Equations - Two Variables

Q 1: Solve for the values of x and y for the following equations: 3x + 6y = 24 and x + 3y = 4
x = -16 and y = 4
x = 6 and y = 8
x = 16 and y = -4

Q 2: Solve for the values of x and y for the following equations: x + y = 5 and x - y = 5
x = 5 and y = 1
x = 5 and y = 0
x = 5 and y = 5

Q 3: Solve for the values of x and y for the following equations: x + y = 6 and x - y = 4
x = 4 and y = 2
x = 5 and y = 1
x = 1 and y = 5

Q 4: Solve for the values of x and y for the following equations: x + y = 10 and 3x + 2y = 24
x = 6 and y = 4
x = 4 and y = 6
x = 2 and y = 3

Q 5: Solve for the values of x and y for the following equations: 2x + y = 6 and 2x + 2y = 24
x = 18 and y = 6
x = 6 and y = -18
x = -6 and y = 18

Q 6: Solve for the values of x and y for the following equations: x + 3y = 7 and 2x + 5y = 12
x = 1 and y = 2
x = 2 and y = 2
x = 1 and y = 1

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Question 8: This question is available to subscribers only!


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