a. The solution set is (-3,2)
b. The solution set is (2,6)
c. The solution set is (1,1)
d. The solution set is (-0.5,1.5)
e. The solution set is (7.5, -4.5)
f. The solution set is (-2,7)
Step-by-step explanation:
a. x+2y=1 Eqn 1
x=y-5 Eqn 2
Putting value of x from Eqn 2 in Eqn 1
![(y-5)+2y=1\\y-5+2y=1\\3y=1+5\\3y=6](https://tex.z-dn.net/?f=%28y-5%29%2B2y%3D1%5C%5Cy-5%2B2y%3D1%5C%5C3y%3D1%2B5%5C%5C3y%3D6)
Dividing both sides by 3
![\frac{3y}{3}=\frac{6}{3}\\y=2](https://tex.z-dn.net/?f=%5Cfrac%7B3y%7D%7B3%7D%3D%5Cfrac%7B6%7D%7B3%7D%5C%5Cy%3D2)
Putting y=2 in Eqn 2
![x=2-5\\x=-3](https://tex.z-dn.net/?f=x%3D2-5%5C%5Cx%3D-3)
The solution set is (-3,2).
b. y-x=4 Eqn 1
y=3x Eqn 2
Putting value of y from Eqn 2 in Eqn 1
![3x-x=4\\2x=4](https://tex.z-dn.net/?f=3x-x%3D4%5C%5C2x%3D4)
Dividing both sides by 2
![\frac{2x}{2}=\frac{4}{2}\\x=2](https://tex.z-dn.net/?f=%5Cfrac%7B2x%7D%7B2%7D%3D%5Cfrac%7B4%7D%7B2%7D%5C%5Cx%3D2)
Putting x=2 in Eqn 2
![y=3(2)\\y=6](https://tex.z-dn.net/?f=y%3D3%282%29%5C%5Cy%3D6)
The solution set is (2,6)
c. y = x Eqn 1
y = -x+2 Eqn 2
Putting value of y from Eqn 1 in Eqn 2
![x=-x+2\\x+x=2\\2x=2](https://tex.z-dn.net/?f=x%3D-x%2B2%5C%5Cx%2Bx%3D2%5C%5C2x%3D2)
Dividing both sides by 2
![\frac{2x}{2}=\frac{2}{2}\\x=1](https://tex.z-dn.net/?f=%5Cfrac%7B2x%7D%7B2%7D%3D%5Cfrac%7B2%7D%7B2%7D%5C%5Cx%3D1)
Putting x=1 in Eqn 1
![y=1](https://tex.z-dn.net/?f=y%3D1)
The solution set is (1,1)
d. y=5x+4 Eqn 1
x+3y=4 Eqn 2
Putting value of y from Eqn 1 in Eqn 2
![x+3(5x+4)=4\\x+15x+12=4\\16x=4-12\\16x=-8](https://tex.z-dn.net/?f=x%2B3%285x%2B4%29%3D4%5C%5Cx%2B15x%2B12%3D4%5C%5C16x%3D4-12%5C%5C16x%3D-8)
Dividing both sides by 16
![\frac{16x}{16}=\frac{-8}{16}\\x=-0.5](https://tex.z-dn.net/?f=%5Cfrac%7B16x%7D%7B16%7D%3D%5Cfrac%7B-8%7D%7B16%7D%5C%5Cx%3D-0.5)
Putting x= -0.5 in Eqn 1
![y=5(-0.5)+4\\y=-2.5+4\\y=1.5](https://tex.z-dn.net/?f=y%3D5%28-0.5%29%2B4%5C%5Cy%3D-2.5%2B4%5C%5Cy%3D1.5)
The solution set is (-0.5,1.5)
e. x-y=12 Eqn 1
y = 3-x Eqn 2
Putting value of y from Eqn 2 in Eqn 1
![x-(3-x)=12\\x-3+x=12\\2x=12+3\\2x=15](https://tex.z-dn.net/?f=x-%283-x%29%3D12%5C%5Cx-3%2Bx%3D12%5C%5C2x%3D12%2B3%5C%5C2x%3D15)
Dividing both sides by 2
![\frac{2x}{2}=\frac{15}{2}\\x=7.5](https://tex.z-dn.net/?f=%5Cfrac%7B2x%7D%7B2%7D%3D%5Cfrac%7B15%7D%7B2%7D%5C%5Cx%3D7.5)
Putting x=7.5 in Eqn 2
![y=3-7.5\\y=-4.5](https://tex.z-dn.net/?f=y%3D3-7.5%5C%5Cy%3D-4.5)
The solution set is (7.5, -4.5)
f. 2x+y=3 Eqn 1
x+y=5 Eqn 2
From Eqn 2;
y = 5-x
Putting in Eqn 1
![2x+(5-x)=3\\2x+5-x=3\\x=3-5\\x=-2](https://tex.z-dn.net/?f=2x%2B%285-x%29%3D3%5C%5C2x%2B5-x%3D3%5C%5Cx%3D3-5%5C%5Cx%3D-2)
Putting x=-2 in Eqn 1
![2(-2)+y=3\\-4+y=3\\y=3+4\\y=7](https://tex.z-dn.net/?f=2%28-2%29%2By%3D3%5C%5C-4%2By%3D3%5C%5Cy%3D3%2B4%5C%5Cy%3D7)
The solution set is (-2,7).
Keywords: linear equation, substitution method
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