Mathematics, 06.12.2020 14:00 neazisa

Solves a systems of linear equations in two variables. Solve the following systems by graphing
1) 2x + y = 5 and x-3y = - 8
2) 6x - 3y = - 9 and 2x + 2y = -6

Solve the following systems by substitution
3) y = 5x-3 and -x - 5y = - 11
4) 2x - 6y = 24 and x - 5y - 22

Solve the following systems by elimination
5) - 4x - 2y = -2 and 4x + 8y = -24
6) x - y = 11 and 2x + y = 19

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Answer from: ari9843

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 $\mathbf{-x - 5y = -11}$

$\mathbf{x= 11 - 5y}$

Substitute $\mathbf{x= 11 - 5y}$ in $\mathbf{y = 5x - 3}$

$\mathbf{y = 5(11 - 5y) - 3}$

Open bracket

$\mathbf{y = 55 - 25y - 3}$

Collect like terms

$\mathbf{y +25y= 55 - 3}$

$\mathbf{26y= 52}$

Divide both sides by 2

$\mathbf{y= 2}$

Substitute $\mathbf{y= 2}$ in $\mathbf{x= 11 - 5y}$

$\mathbf{x =11 - 5(2)}$

$\mathbf{x =1}$

So, the solution is (x,y) = (1,2)

4) 2x - 6y = 24 and x - 5y = 22

Make x the subject in $\mathbf{x - 5y = 22}$

$\mathbf{x = 5y + 22}$

Substitute $\mathbf{x = 5y + 22}$ in $\mathbf{2x - 6y =24}$

$\mathbf{2(5y + 22) - 6y =24}$

$\mathbf{10y + 44 - 6y =24}$

Collect like terms

$\mathbf{10y - 6y =24 - 44}$

$\mathbf{4y =- 20}$

Divide by 4

$\mathbf{y =- 5}$

Substitute $\mathbf{y =- 5}$ in $\mathbf{x = 5y + 22}$

$\mathbf{x = 5(-5) + 22}$

$\mathbf{x = -3}$

So, the solution is (x,y) = (-3,-5)

Elimination

5) - 4x - 2y = -2 and 4x + 8y = -24

Add both equations to eliminate x

$\mathbf{-4x + 4x - 2y + 8y = -2- 24}$

$\mathbf{6y = -26}$

Divide both sides by 6

$\mathbf{y = -\frac{13}{3}}$

Substitute $\mathbf{y = -\frac{13}{3}}$ in $\mathbf{4x + 8y = -24}$

$\mathbf{4x - 8 \times \frac{13}{3} = -24}$

$\mathbf{4x - \frac{104}{3} = -24}$

Collect like terms

$\mathbf{4x = \frac{104}{3} -24}$

$\mathbf{4x = \frac{104-72}{3} }$

$\mathbf{4x = \frac{32}{3} }$

Divide both sides by 4

$\mathbf{x = \frac{8}{3} }$

Hence, the solution is (x,y) = (8/3,-13/3)

6) x - y = 11 and 2x + y = 19

Add both equations to eliminate y

$\mathbf{x + 2x - y + y = 11 +19}$

$\mathbf{3x = 30}$

Divide through by 3

$\mathbf{x = 10}$

Substitute $\mathbf{x = 10}$ in $\mathbf{x - y = 11}$

$\mathbf{10 - y = 11}$

Collect like terms

$\mathbf{ y =10 - 11}$

$\mathbf{ y =- 1}$

Hence, the solution is (x,y) = (10,-1)

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