If the two straight lines 𝐿 one
negative eight 𝑥 plus seven 𝑦 minus nine equals zero and 𝐿 two 𝑎𝑥 plus 24𝑦
plus 56 equals zero are perpendicular, find the value of 𝑎.
We can go ahead and copy down the
two equations. We know that the slopes of two
perpendicular lines multiplying together to equal zero, they are the negative
reciprocal of one another. But before we can start thinking
about reciprocals, we’ll need to first find the slope of both of these lines.
Starting with 𝐿 one, I wanna
convert this equation to slope-intercept form. That’s a format that’s in 𝑦 on the
left and everything else on the right. To do that, we move the constant to
the right side by adding nine. Negative nine plus nine equals
zero. Zero plus nine equals nine. Now we have a negative eight 𝑥
plus seven 𝑦. I’ll add eight 𝑥 to the left and
the right. Negative eight 𝑥 plus positive
eight 𝑥 equals zero. And on the right, we now have eight
𝑥 plus nine.
I still want my 𝑦 by itself. So I can divide seven 𝑦 by seven,
leaving me with 𝑦. If I divide by seven on the left, I
need to divide by seven on the right. The right side now says eight 𝑥
over seven plus nine over seven. I can pull out the slope of line
one. The slope of line one is
eight-sevenths. I wanna follow the same process and
get line two into that 𝑦 equals format.
First I’ll get the constant to the
other side, by subtracting 56. That leaves us with 𝑎𝑥 plus 24𝑦
on the left and negative 56 on the right. Now I want to move the 𝑥 value to
the right-hand side. I subtract 𝑎𝑥 from both sides of
the equation. 𝑎𝑥 minus 𝑎𝑥 cancels out,
leaving us with 24𝑦 equals negative 56 minus 𝑎𝑥. Again, we’re going for that 𝑦
equals. So I need to divide the left by
If I divide one side by 24 I need
to divide the other side by 24. Now we see that 𝑦 equals negative
56 over 24 minus 𝑎𝑥 over 24. The slope of this equation is
negative 𝑎 over 24. Now we can go back to this idea of
the slopes being the negative reciprocals of one another. The slope of line one equals eight
over seven. The negative reciprocal of eight
over seven equals negative seven-eighths.
And that means we wanna take the
slope of line two and set that equal to negative seven-eighths, like this: negative
𝑎 over 24 is equal to negative seven over eight. To solve for 𝑎, I’m going to
multiply the left side by negative 24. And if I multiply the left side by
something, I need to multiply the right side by the same amount. On the left, everything cancels out
leaving us with only 𝑎.
Negative seven times negative 24 on
the right-hand side equals 168 all over eight. And 168 divided by eight equals
21. If we want these two lines to be
perpendicular, the value of 𝑎 will be 21.