18JUL21

SOLVING LINEAR EQUATIONS IN TWO VARIABLES


Example 1:

x+y=5

x-y=1

Where do these two lines intersect?


Let's add the two equations.

(x+y)+(x-y)=5+1

x+y+x-y=6

2x=6

x=6/2=3

x+y=5

3+y=5

y=5-3=2

(3,2) is the point of intersection of the given lines.


Example 2:

2x+3y=14

5x-y=1

Where do the two lines cut?

2x+3y+5x-y=14+1

7x+2y=15

This isn't working.

Again, let's try differently.

2x+3y=14

5x-y=1

Let's multiply both sides of this equation by 3.

15x-3y=3

Adding the two equations,

2x+3y+15x-3y=14+3

17x=17

x=17/17=1

5x-y=1

5(1)-y=1

5-y=1

-y=1-5

-y=-4

y=4

(1,4) is the point of intersection of the given lines.


Example 3:

2x+5y+29=0

3x+2y+16=0

Let's multiply the first equation by 2 and the second equation by 5.

4x+10y+58=0

15x+10y+80=0

Subtracting the second equation from the first one,

(4x+10y+58)-(15x+10y+80)=0

4x+10y+58-15x-10y-80=0

4x-15x+58-80=0

-11x-22=0

-11x=22

x==22/-11=-22/11=-2

2x+5y+29=0

2(-2)+5y+29=0

-4+5y+29=0

5y+25=0

5y=-25

y=-25/5=-5

(-2,-5) is the point of intersection of the given lines.


Example 4:

2x+y=5

4x+2y=9

Multiplying the first equation by 2,

4x+2y=10

4x+2y=9

Subtracting the second equation from the first one,

(4x+2y)-(4x+2y)=10-9

4x+2y-4x-2y=1

0=-1

This doesn't make sense.

They don't intersect.

THEY ARE PARALLEL.


2x+y=5

y=-2x+5⇒slope = -2


4x+2y=9

2y=-4x+9

y=(-4x+9)/2

y=(-4x/2)+(9/2)

y=(-4/2)x+(9/2)

y=-2x+(9/2)⇒slope = -2

Since the slope of the two lines is the same, they are parallel.

OBSERVATION: A PAIR OF LINEAR EQUATIONS WILL NOT HAVE ANY SOLUTION IF THE SLOPES ARE EQUAL. A SOLUTION EXISTS ONLY IF THE SLOPES ARE DIFFERENT.


The Relation between Dividend and Divisor
When 7 is divided by 2,
2 divides 7 three times and 1 is left.
Here, 7 is Dividend (D), 2 is Divisor (d), 3 is Quotient (q) and 1 is Remainder (r).
7 = 3*2 + 1
D = q*d + r
Let's compare this to
y = m*x + c
y is analogous to dividend, x is analogous to divisor, slope (m) is analogous to quotient and y-intercept (c) is analogous to remainder.
In the above example, x=2 and y=7.

If you imagine a line passing through (2,7) and making a y-intercept of 1, its slope will be "3".


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