Linear equations are represented by straight lines.
Graphs
(1) 2x + y = 5 and x - 3y = -8
See attachment for the graphs of 2x + y = 5 and x - 3y = -8
From the graph, we have:
(x,y) = (1,3)
(2) 6x - 3y = -9 and 2x + 2y = -6
See attachment for the graphs of 6x - 3y = -9 and 2x + 2y = -6
From the graph, we have:
(x,y) = (-2,-1)
Substitution
3) y = 5x-3 and -x - 5y = - 11
Make x the subject in [tex]\mathbf{-x - 5y = -11}[/tex]
[tex]\mathbf{x= 11 - 5y}[/tex]
Substitute [tex]\mathbf{x= 11 - 5y}[/tex] in [tex]\mathbf{y = 5x - 3}[/tex]
[tex]\mathbf{y = 5(11 - 5y) - 3}[/tex]
Open bracket
[tex]\mathbf{y = 55 - 25y - 3}[/tex]
Collect like terms
[tex]\mathbf{y +25y= 55 - 3}[/tex]
[tex]\mathbf{26y= 52}[/tex]
Divide both sides by 2
[tex]\mathbf{y= 2}[/tex]
Substitute [tex]\mathbf{y= 2}[/tex] in [tex]\mathbf{x= 11 - 5y}[/tex]
[tex]\mathbf{x =11 - 5(2)}[/tex]
[tex]\mathbf{x =1}[/tex]
So, the solution is (x,y) = (1,2)
4) 2x - 6y = 24 and x - 5y = 22
Make x the subject in [tex]\mathbf{x - 5y = 22}[/tex]
[tex]\mathbf{x = 5y + 22}[/tex]
Substitute [tex]\mathbf{x = 5y + 22}[/tex] in [tex]\mathbf{2x - 6y =24}[/tex]
[tex]\mathbf{2(5y + 22) - 6y =24}[/tex]
[tex]\mathbf{10y + 44 - 6y =24}[/tex]
Collect like terms
[tex]\mathbf{10y - 6y =24 - 44}[/tex]
[tex]\mathbf{4y =- 20}[/tex]
Divide by 4
[tex]\mathbf{y =- 5}[/tex]
Substitute [tex]\mathbf{y =- 5}[/tex] in [tex]\mathbf{x = 5y + 22}[/tex]
[tex]\mathbf{x = 5(-5) + 22}[/tex]
[tex]\mathbf{x = -3}[/tex]
So, the solution is (x,y) = (-3,-5)
Elimination
5) - 4x - 2y = -2 and 4x + 8y = -24
Add both equations to eliminate x
[tex]\mathbf{-4x + 4x - 2y + 8y = -2- 24}[/tex]
[tex]\mathbf{6y = -26}[/tex]
Divide both sides by 6
[tex]\mathbf{y = -\frac{13}{3}}[/tex]
Substitute [tex]\mathbf{y = -\frac{13}{3}}[/tex] in [tex]\mathbf{4x + 8y = -24}[/tex]
[tex]\mathbf{4x - 8 \times \frac{13}{3} = -24}[/tex]
[tex]\mathbf{4x - \frac{104}{3} = -24}[/tex]
Collect like terms
[tex]\mathbf{4x = \frac{104}{3} -24}[/tex]
[tex]\mathbf{4x = \frac{104-72}{3} }[/tex]
[tex]\mathbf{4x = \frac{32}{3} }[/tex]
Divide both sides by 4
[tex]\mathbf{x = \frac{8}{3} }[/tex]
Hence, the solution is (x,y) = (8/3,-13/3)
6) x - y = 11 and 2x + y = 19
Add both equations to eliminate y
[tex]\mathbf{x + 2x - y + y = 11 +19}[/tex]
[tex]\mathbf{3x = 30}[/tex]
Divide through by 3
[tex]\mathbf{x = 10}[/tex]
Substitute [tex]\mathbf{x = 10}[/tex] in [tex]\mathbf{x - y = 11}[/tex]
[tex]\mathbf{10 - y = 11}[/tex]
Collect like terms
[tex]\mathbf{ y =10 - 11}[/tex]
[tex]\mathbf{ y =- 1}[/tex]
Hence, the solution is (x,y) = (10,-1)
Read more about linear equations at:
https://brainly.com/question/11897796
