Question Video: Finding the Values of the Unknown Coefficients in Two Parallel Straight Lines’ Equations

The straight lines 8𝑥 + 5𝑦 = 8 and 8𝑥 + 𝑎𝑦 = −8 are parallel. What is the value of 𝑎?

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Video Transcript

The straight lines eight 𝑥 plus five 𝑦 equals eight and eight 𝑥 plus 𝑎𝑦 equals negative eight are parallel. What is the value of 𝑎?

We know that parallel lines have the same slope. And in slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏, the coefficient of the 𝑥-variable 𝑚 represents the slope. We’ve been told that these two lines are parallel. And that means they will have the same slope. To find the slope of these lines, we’ll convert them to slope–intercept form. To do that, we get 𝑦 by itself. Since both equations have eight 𝑥 on the left, we’ll subtract eight 𝑥 from both sides of both equations. On the left, we would have five 𝑦 equals negative eight 𝑥 plus eight. And on the right, 𝑎𝑦 equals negative eight 𝑥 minus eight. We need to get 𝑦 by itself for slope–intercept form. So, we can divide through by five. And the equation on the left in slope–intercept form will be 𝑦 equals negative eight-fifths 𝑥 plus eight-fifths.

On the right, we need to do something similar. To get 𝑦 by itself, we’ll divide through by 𝑎. And our second equation will be 𝑦 equals negative eight over 𝑎𝑥 minus eight over 𝑎. These slopes have to be equal to each other if these lines are parallel. Since one of the slopes is negative eight over five, the other slope will need to be negative eight over five. And that tells us that 𝑎 must be five for these two lines to be parallel. If we go back in and plug in five for 𝑎, we see that the ratio of coefficients between these two equations are equal to each other, which makes them parallel.

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