Video: Using a Given Relation between Two Straight Lines to Solve a Problem

If the two straight lines 𝐿₁: βˆ’8π‘₯ + 7𝑦 βˆ’ 9 = 0 and 𝐿₂: π‘Žπ‘₯ + 24𝑦 + 56 = 0 are perpendicular, find the value of π‘Ž.

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Video Transcript

If the two straight lines 𝐿 one negative eight π‘₯ plus seven 𝑦 minus nine equals zero and 𝐿 two π‘Žπ‘₯ plus 24𝑦 plus 56 equals zero are perpendicular, find the value of π‘Ž.

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 π‘₯ plus seven 𝑦. I’ll add eight π‘₯ to the left and the right. Negative eight π‘₯ plus positive eight π‘₯ equals zero. And on the right, we now have eight π‘₯ 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 π‘₯ 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 π‘Žπ‘₯ plus 24𝑦 on the left and negative 56 on the right. Now I want to move the π‘₯ value to the right-hand side. I subtract π‘Žπ‘₯ from both sides of the equation. π‘Žπ‘₯ minus π‘Žπ‘₯ cancels out, leaving us with 24𝑦 equals negative 56 minus π‘Žπ‘₯. Again, we’re going for that 𝑦 equals. So I need to divide the left by 24.

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 π‘Žπ‘₯ over 24. The slope of this equation is negative π‘Ž 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 π‘Ž over 24 is equal to negative seven over eight. To solve for π‘Ž, 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 π‘Ž.

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 π‘Ž will be 21.

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