Question

worth 35 points 1-8 Determine if the following equations have one solution, no soution, or infinite solution

Answer 1

A linear equation has infinite solutions if the left and right sides are the same. These equations can be manipulated into the expression 0=0, which is true no matter which value we choose for x, so they have infinite solution.

It has no solutions if the terms involving x cancel out, but the constant terms are different. These equations can be turned into an expression like
`a=b`, where a and b are actually different numbers, and so it can never be true.

In all other cases, you have exactly one solution.

1

`2x+2x+2=4x+2 \iff 4x+2=4x+2 \iff` infinite solutions

2

`3(x-1)=2x+9 \iff 3x-3 = 2x+9 \iff x=12`

3

`2x+8=2(x+4) \iff 2x+8=2x+8\iff` infinite solutions

4

`8(x+2)=2x+16 \iff 8x+16=2x+16 \iff 6x=0 \iff x=0`

5

`4x+1 = 2(2x+3) \iff 4x+1 = 4x+6 \iff 1=6 \iff` no solutions

6

`-2(x+1) = 2(x-1) \iff -2x-2=2x-2 \iff 4x=0 \iff x=0`

7

`3x+1=3(x-1)+4 \iff 3x+1=3x-3+4 \iff 3x+1=3x+1\iff` infinite solutions

8

`(1)/(2)(2-4x)+2x=13 \iff 1-2x+2x=13 \iff 1=13 \iff` no solutions.

Answer 2

A linear equation has:

• Infinite solutions if the left and right sides are the same.
• No solutions if the terms involving x cancel out, but the constant terms are different.
• Other than above cases, you have exactly one solution.

Explanation:

Part 1:

Equation: 2x + 2x + 2 = 4x + 2

Solution:

2x + 2x + 2 = 4x + 2

=4x + 2 = 4x + 2 Infinite solution.

=4x - 4x = 2-2

=0 = 0

Part 2.

Equation: 3(x - 1) = 2x + 9

Solution:

= 3(x - 1) = 2x + 9

=3x - 3 = 2x + 9

=3x - 2x = 9 + 3

=x = 12 One solution.

Part 3.

Equation: 2x + 8 =2(x + 4)

Solution:

2x + 8 = 2x + 8 Infinite solution.

Part 4.

Equation: 8(x+2) = 2x + 16

Solution:

= 8x + 16 = 2x +16

= 8x - 2x = 16 - 16

= 6x = 0

= x = 0 One solution.

Part 5.

Equation: 4x + 1 = 2(2x + 3)

Solution:

4x + 1 = 4x + 6

= 0 = 5 No solution.

Part 6.

Equation: -2(x+1) = 2(x-1)

Solution:

= -2x - 2 = 2x - 2

= -4x = 0

= x = 0/-4 = 0 One solution.

Part 7.

Equation: 3x + 1 = 3( x- 1) + 4

Solution:

=3x +1 = 3x - 3 +4

= 3x + 1 = 3x +1 Infinite solution.

Part 8.

Equation: 1/2 (2 - 4x) +2x = 13

Solution:

= 1 - 2x + 2x = 13

= 1 = 13 No solution.