2. Determine the parameter $a$ so that the lines with equations $(a-1) x+a y-5=0$ and $a x+(3 a-1) y-7=0$ intersect on the x-axis. Write the coordinates of the intersection point of the lines.
2. We note that the ordinate of the intersection is equal to 0. In the equations of the lines, we substitute $y=0$:

$(a-1) x-5=0$ and $a x-7=0$. From the second equation, we express $x=\frac{7}{a}$ and substitute it into the first. We get $(a-1) \frac{7}{a}-5=0$, eliminate the fraction, simplify $7 a-7-5 a=0$. We calculate $a=\frac{7}{2}$, which we substitute into one of the line equations: for example, $\frac{7}{2} x-7=0$. We calculate $x=2$. We write the coordinates of the intersection $P(2,0)$.

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-08.jpg?height=66&width=1642&top_left_y=2097&top_left_x=207)

Solving the system of equations $(a-1) x-5=0$ and $a x-7=0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 2^{*}$ at the point

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-08.jpg?height=63&width=1630&top_left_y=2199&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-08.jpg?height=49&width=1627&top_left_y=2260&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-08.jpg?height=60&width=1625&top_left_y=2306&top_left_x=210)