Find the number of integer values of m such t | vedclass

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(A) Given the equations: $3x + 4y = 9$ $y = mx + 1$ Substitute $y$ in the first equation: $3x + 4(mx + 1) = 9$ $3x + 4mx + 4 = 9$ $x(3 + 4m) = 5$ $x =

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$2$

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(A) Given the equations: $3x + 4y = 9$ $y = mx + 1$ Substitute $y$ in the first equation: $3x + 4(mx + 1) = 9$ $3x + 4mx + 4 = 9$ $x(3 + 4m) = 5$ $x = \frac{5}{3 + 4m}$ For $x$ to be an integer,$(3 + 4m)$ must be a divisor of $5$. The divisors of $5$ are $\pm 1$ and $\pm 5$. Case $1$: $3 + 4m = 1 \implies 4m = -2 \implies m = -0.5$ (Not an integer) Case $2$: $3 + 4m = -1 \implies 4m = -4 \implies m = -1$ (Integer) Case $3$: $3 + 4m = 5 \implies 4m = 2 \implies m = 0.5$ (Not an integer) Case $4$: $3 + 4m = -5 \implies 4m = -8 \implies m = -2$ (Integer) Thus,the integer values of $m$ are $-1$ and $-2$. There are $2$ such integer values.

Find the number of integer values of $m$ such that the $x$-coordinate of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is an integer.

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