Question

Solve each system of equations by showing the process of substitution.

a) y=-1/4 x and x+2y=4
b) y=-x-2 and 3x+3y=6
c) -8x+2y=4 and y=4x+2

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Answer 1

A. {x,y}={-2,-3}

// Solve equation [2] for the variable x

[2] x = 2y + 4

// Plug this in for variable x in equation [1]

[1] (2y+4) - y = 1

[1] y = -3

// Solve equation [1] for the variable y

[1] y = - 3

// By now we know this much :

x = 2y+4

y = -3

// Use the y value to solve for x

x = 2(-3)+4 = -2

B. [1] 3x=3y-6

[2] y=x+2

Equations Simplified or Rearranged :

[1] 3x - 3y = -6

[2] -x + y = 2

Solve by Substitution :

// Solve equation [2] for the variable y

[2] y = x + 2

// Plug this in for variable y in equation [1]

[1] 3x - 3•(x +2) = -6

[1] 0 = 0 => Infinitely many solutions

C.Step by Step Solution

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System of Linear Equations entered :

[1] 4x - y = 2

[2] 8x - 2y = 4

Solve by Substitution :

// Solve equation [1] for the variable y

[1] y = 4x - 2

// Plug this in for variable y in equation [2]

[2] 8x - 2•(4x-2) = 4

[2] 0 = 0 => Infinitely many solutions

Answer 2

Explanation:

a) y = -1/4x --------------(I)

x + 2y = 4 -----------------(II)

Substitute y = (-1/4)x in equation (I)

`x + 2*(-1)/(4)x=4\\\\\\x -(1)/(2)x=4\\\\Multiply \ the \ entire \ equation \ by \ 2 \\\\2x -x = 8\\\\x=8`

`Substitute \ x = 8 \ in \ equation \ (I)\\\\y=(-1)/(4)*8\\\\\\y = -2`

Answer: x = 8 ; y = -2

b) y = -x - 2 --------------(i)

3x + 3y = 6 -----------(ii)

Divide equation (ii) by m

x + y = 2

y = -x + 2 ----(iii)

From (i) and (iii), it shows that these lines have same slope. So, they are parallel lines

Answer: No solution

3) -8x + 2y =4

2y = 8x + 4

Divide the entire equation by 2

y = 4x + 2 -------------(i)

y = 4x + 2 ----------------(ii)

From (i) and (ii), we come to know that these lines coincide.So, they have infinite solutions.