We know that:
• If the two lines have different slopes, the system has exactly one solution.
,
• If the two lines have the same slope and y-intercept, the system has infinite solutions.
,
• If the two lines have the same slope and different y-intercepts, they are parallel, and the system has no solutions.
Then, we need to know the slopes of the lines.
• Line 1
We write the equation in its slope-intercept form. For this, we solve the equation for y.
[tex]\begin{gathered} y=mx+b\Rightarrow\text{ Slope}-\text{intercept form} \\ \text{ Where m is the slope and} \\ b\text{ is the y-intercept} \end{gathered}[/tex][tex]\begin{gathered} -8x+9y=-8 \\ \text{ Add 8x from both sides} \\ -8x+9y+8x=-8+8x \\ 9y=-8+8x \\ \text{ Divide by 9 from both sides} \\ \frac{9y}{9}=\frac{-8+8x}{9} \\ y=-\frac{8}{9}+\frac{8}{9}x \\ \text{ Reorder} \\ y=\frac{8}{9}x-\frac{8}{9} \end{gathered}[/tex]
Then, the slope of this line is 8/9.
• Line 2
As we can see, this line is already in its slope-intercept form.
Then, the slope of this line is -6/7.
Since the lines have different slopes, the system has exactly one solution.
