HELPPPPP
Which linear equations have an infinite number of solutions? Check all that apply.
(x – ) = (x – left-parenthesis x minus StartFraction 3 Over 7 EndFraction right-parenthesis equals StartFraction 2 Over 3 EndFraction left-parenthesis StartFraction 3 Over 2 x EndFraction minus StartFraction 9 Over 14 EndFraction right-parenthesis.)
8(x + 2) = 5x – 14
12.3x – 18 = 3(–6 + 4.1x)
(6x + 10) = 7(StartFraction one-half EndFraction left parenthesis 6 x plus 10 right-parenthesis equals 7 left-parenthesis StartFraction 3 Over 7 x minus 2 right-parenthesis.x – 2)
4.2x – 3.5 = 2.1 (5x + 8)

First equation solves to -3/7=-18/42...take a 6 from 18 and 42, and you get -3/7. -3/7=-3/7 which is true - that means that, no matter what value is inserted for x, a true equation will result = infinite number of solutions

Second equation simplifies to 3x=-30. Nope

Third equation simplifies to -18=-18 = infinite number of solutions

Fourth eq: 5=-7/2 = No solution

5th eq: 6.2x=-3.5-16.2 = nope

MrRoyal MrRoyal · Answer 2 · 2021-10-03T07:03:10+02:00

An equation with an infinite number of solutions, means that the variable can take any value, and the equation will still be true

The equations with an infinite number of solutions are:

[tex]\mathbf{(x - \frac 37) = \frac 23(\frac 32x - \frac 9{14})}[/tex]
[tex]\mathbf{12.3x - 18 = 3(-6 + 4.1x)}[/tex]

Start by testing the options

(a)

[tex]\mathbf{(x - \frac 37) = \frac 23(\frac 32x - \frac 9{14})}[/tex]

Open brackets

[tex]\mathbf{x - \frac 37 = x - \frac 37}[/tex]

Subtract x from both sides

[tex]\mathbf{- \frac 37 = - \frac 37}[/tex]

The above equation is true.

This means that [tex]\mathbf{(x - \frac 37) = \frac 23(\frac 32x - \frac 9{14})}[/tex] has an infinite number of solutions

(b)

[tex]\mathbf{ 8(x+ 2) = 5x -14}[/tex]

Open bracket

[tex]\mathbf{8x + 16 = 5x -14}[/tex]

Collect like terms

[tex]\mathbf{8x -5x=- 16 -14}[/tex]

[tex]\mathbf{3x=- 30}[/tex]

Divide both sides by 3

[tex]\mathbf{x=- 10}[/tex]

The solution to [tex]\mathbf{ 8(x+ 2) = 5x -14}[/tex] is [tex]\mathbf{-10}[/tex]

(c)

[tex]\mathbf{12.3x - 18 = 3(-6 + 4.1x)}[/tex]

Open brackets

[tex]\mathbf{12.3x - 18 = -18 + 12.3x}[/tex]

Subtract 12.3x from both sides

[tex]\mathbf{ - 18 = -18}[/tex]

The above equation is true.

This means that [tex]\mathbf{12.3x - 18 = 3(-6 + 4.1x)}[/tex] has an infinite number of solutions

(d)

[tex]\mathbf{6x + 10 = 7\frac 12(\frac 37x - 2)}[/tex]

Express fractions as improper fractions

[tex]\mathbf{6x + 10 = \frac{15}{2}(\frac 37x - 2)}[/tex]

Multiply through by 2

[tex]\mathbf{12x + 20 = 15(\frac 37x - 2)}[/tex]

Multiply through by 7

[tex]\mathbf{84x + 140 = 210(\frac 37x - 2)}[/tex]

[tex]\mathbf{84x + 140 = 90x - 410}[/tex]

Collect like terms

[tex]\mathbf{90x - 84x = 140 + 410}[/tex]

[tex]\mathbf{6x = 550}[/tex]

Divide through by 6

[tex]\mathbf{x = 91\frac 23}[/tex]

The solution to [tex]\mathbf{6x + 10 = 7\frac 12(\frac 37x - 2)}[/tex] is [tex]\mathbf{91\frac 23}[/tex]

(e)

[tex]\mathbf{4.2x - 3.5 = 2.1(5x + 8)}[/tex]

Open brackets

[tex]\mathbf{4.2x - 3.5 = 10.5x + 16.8}[/tex]

Collect like terms

[tex]\mathbf{4.2x -10.5x = 3.5 + 16.8}[/tex]

[tex]\mathbf{-6.3x = 20.3}[/tex]

Solve for x

[tex]\mathbf{x = -3.22}[/tex]

The solution to [tex]\mathbf{4.2x - 3.5 = 2.1(5x + 8)}[/tex] is [tex]\mathbf{-3.22}[/tex]

Hence, the equations with an infinite number of solutions are:

[tex]\mathbf{(x - \frac 37) = \frac 23(\frac 32x - \frac 9{14})}[/tex] and [tex]\mathbf{12.3x - 18 = 3(-6 + 4.1x)}[/tex]