Question

Without graphing determine the number of solutions to the system of equations

Answer 1

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.

`\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}`
`\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} \\ (9y)/(9)=(-8+8x)/(9) \\ y=-(8)/(9)+(8)/(9)x \\ \text{ Reorder} \\ y=(8)/(9)x-(8)/(9) \end{gathered}`

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.