Question

which linear equations have an infinite number of solutions? check all that apply.
$\left(x - \frac{3}{7}\
ight) = \frac{2}{3}\left(\frac{3}{2}x - \frac{9}{14}\
ight)$
$8(x + 2) = 5x - 14$
$12.3x - 18 = 3(-6 + 4.1x)$
$\frac{1}{2}(6x + 10) = 7\left(\frac{3}{7}x - 2\
ight)$
$4.2x - 3.5 = 2.1(5x + 8)$

Explanation:

Step1: Simplify first equation

Expand right-hand side:

$$\begin{align*} x - \frac{3}{7} &= \frac{2}{3} \times \frac{3}{2}x - \frac{2}{3} \times \frac{9}{14} \\ x - \frac{3}{7} &= x - \frac{3}{7} \end{align*}$$

Step2: Simplify second equation

Expand and rearrange terms:

$$\begin{align*} 8x + 16 &= 5x - 14 \\ 8x - 5x &= -14 - 16 \\ 3x &= -30 \end{align*}$$

Step3: Simplify third equation

Expand right-hand side:

$$\begin{align*} 12.3x - 18 &= 3\times(-6) + 3\times4.1x \\ 12.3x - 18 &= -18 + 12.3x \\ 12.3x - 18 &= 12.3x - 18 \end{align*}$$

Step4: Simplify fourth equation

Expand both sides:

$$\begin{align*} \frac{1}{2}\times6x + \frac{1}{2}\times10 &= 7\times\frac{3}{7}x - 7\times2 \\ 3x + 5 &= 3x - 14 \\ 3x - 3x &= -14 -5 \\ 0 &= -19 \end{align*}$$

Step5: Simplify fifth equation

Expand right-hand side:

$$\begin{align*} 4.2x - 3.5 &= 2.1\times5x + 2.1\times8 \\ 4.2x - 3.5 &= 10.5x + 16.8 \\ 4.2x - 10.5x &= 16.8 + 3.5 \\ -6.3x &= 20.3 \end{align*}$$

Answer:

$\boldsymbol{(x - \frac{3}{7}) = \frac{2}{3}(\frac{3}{2}x - \frac{9}{14})}$
$\boldsymbol{12.3x - 18 = 3(-6 + 4.1x)}$