The system of linear equations $kx + 5y = 5$, $2x + 3y = 5$ will be consistent if

For the system of linear equations $kx + 5y = 5$ and $2x + 3y = 5$ to be consistent, it must have at least one solution. A system is inconsistent when the lines are parallel but distinct (no solution).

For two equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, the system is inconsistent when:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

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From the given equations:

$a_1 = k$, $b_1 = 5$, $c_1 = 5$

$a_2 = 2$, $b_2 = 3$, $c_2 = 5$

The ratios are:

$\frac{a_1}{a_2} = \frac{k}{2}$

$\frac{b_1}{b_2} = \frac{5}{3}$

$\frac{c_1}{c_2} = \frac{5}{5} = 1$

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The system is inconsistent when:

$\frac{k}{2} = \frac{5}{3}$ and $\frac{k}{2} \neq 1$

$\frac{k}{2} = \frac{5}{3}$

$k = \frac{10}{3}$

Since $\frac{10/3}{2} = \frac{5}{3} \neq 1$, the condition is satisfied.

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The system is consistent for all values of $k$ except when the lines are parallel and distinct.

Therefore, the system is consistent when $k \neq \frac{10}{3}$