Express the equation of the line passing through points $(1,7)$ and $(2,-3)$ in the form $ax+by=c$, where $a$, $b$ and $c$ are integers, the greatest common divisor of $a$ and $b$ is $1,$ and $c<0$.
Using the two-point form, the equation of the line passing through $(1,7)$ and $(2,-3)$ is \[(y - 7) = \left(\frac{7 - (-3)}{1 - 2}\right)(x - 1).\]
Simplifying, we have \[y - 7 = 10(x - 1).\] Expanding the product on the right and rearranging the terms, we obtain \[10x - y = -3.\] Thus, the equation of the line in the desired form is $\boxed{10x - y = -3}$.
To find the equation of the line passing through the points $(1,7)$ and $(2,-3)$, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Step 1: Find the slope (m):
The slope, $m$, is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1},\] where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two given points.
Using the formula, we get:
\[m = \frac{-3 - 7}{2 - 1} = \frac{-10}{1} = -10.\]
Step 2: Find the y-intercept (b):
We can find the y-intercept, $b$, by substituting the coordinates of one of the points into the equation $y = mx + b$ and solving for $b$.
Let's use the point $(1,7)$. Plugging in the values, we get:
\[7 = -10 \cdot 1 + b.\]
Simplifying the equation, we have:
\[b = 7 + 10 = 17.\]
Step 3: Write the equation in the required form $ax + by = c$:
Substituting the slope ($m = -10$) and the y-intercept ($b = 17$) into the equation, we have:
\[y = -10x + 17.\]
To write the equation in the required form $ax + by = c$, we can rearrange the equation as follows:
\[10x + y = 17.\]
Therefore, the equation of the line passing through the points $(1,7)$ and $(2,-3)$ in the form $ax + by = c$, where $a$, $b$ and $c$ are integers, the greatest common divisor of $a$ and $b$ is $1$, and $c < 0$, is:
\[10x + y = 17.\]
To find the equation of the line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, we can use the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$, where $m$ is the slope of the line.
First, let's find the slope $m$ of the line passing through $(1, 7)$ and $(2, -3)$. The slope is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the given values, we get:
\[m = \frac{-3 - 7}{2 - 1} = \frac{-10}{1} = -10\]
Now that we have the slope, we can use the point-slope form of the linear equation and one of the given points to determine the equation of the line. Let's use the point $(1, 7)$. Plugging in the values, we have:
\[y - 7 = -10(x - 1)\]
Simplifying, we get:
\[y - 7 = -10x + 10 \]
To express the equation in the form $ax + by = c$, we need to rearrange the equation:
\[10x + y = 17\]
To satisfy the conditions, we can multiply both sides by $-1$, which gives us:
\[-10x - y = -17\]
Therefore, the equation of the line passing through the points $(1, 7)$ and $(2, -3)$ in the desired form is $-10x - y = -17$.