In the 𝑥𝑦-plane, which of the following equations has a graph that is perpendicular to the graph of the equation negative three 𝑥 plus five 𝑦 equals eight. A) Three 𝑥 plus five 𝑦 equals eight. B) Five 𝑥 plus three 𝑦 equals eight. C) Three 𝑥 plus eight 𝑦 equals five. Or D) five 𝑥 minus three 𝑦 equals eight.
So, here, we have this equation. And it’s saying that if we were to graph this equation which of these equations, if they were graphed, would be perpendicular to our original equation. To figure this out, we must look at their slopes. And with the equation of a line 𝑦 equals 𝑚𝑥 plus 𝑏, 𝑚 is the slope. And slope is the vertical change divided by the horizontal change, essentially the steepness of the graph.
So, we wanna know which of these equations, if graphed, would be perpendicular to the graph of our equation. So, the slope will have to be a negative reciprocal of our original slope. But first, our original equation negative three 𝑥 plus five 𝑦 equals eight is not in the correct form for the equation of the line in order to know what the slope is. So, we need to solve for 𝑦.
So, we add three 𝑥 to both sides. On the left, the three 𝑥s cancel. And on the right, we cannot combine three 𝑥 and eight, so we leave it as five 𝑦 equals three 𝑥 plus eight. And now, we divide by five. So, the equation of our line would be 𝑦 equals three-fifths 𝑥 plus eight-fifths. So, three-fifths must be our slope. So, we want the reciprocal and we want to change the sign of that.
So, the negative reciprocal doesn’t mean that it needs to be negative. It just needs to be the opposite sign. So, if it’s a positive three-fifths, we need to make it a negative. And then, reciprocal means we need to flip the fraction. So, the negative reciprocal of three-fifths would be negative five-thirds. Unfortunately, all of our answers are not in the correct form either. So, we need to go through each one and figure out which one would have negative five-thirds as our slope.
Let’s begin with A. To solve for 𝑦, we would be subtracting three 𝑥 from both sides. And then dividing by five, we have that 𝑦 is equal to negative three-fifths 𝑥 plus eight-fifths. So, the slope here would be negative three-fifths, which is not the right slope. For option B, we will begin by subtracting five 𝑥 from both sides then dividing by three. And here, our slope is negative five-thirds. So, option B is probably our answer. But let’s make sure C and D do not work.
For option C, we would first subtract three 𝑥 from both sides and then divide by eight, creating 𝑦 equals negative three-eighths 𝑥 plus five-eighths. Negative three-eighths is not the correct slope, so we can eliminate option C.
Now, lastly, for option D, we subtract five 𝑥 from both sides then divided by negative three. So, the negative threes cancel with the 𝑦. And then, negative five 𝑥 over negative three creates five-thirds 𝑥 because the two negatives create a positive. And then, lastly, we have positive eight over negative three. So, we can just write it as minus eight-thirds.
So, our slope here is five-thirds. However, we wanted the negative reciprocal. So, we needed negative five-thirds. And again, negative doesn’t always mean it has to be negative. But it is in this case, however, negative. So, we can eliminate option D. So, if we were to graph five 𝑥 plus three 𝑦 equals eight, it would be perpendicular to the graph of our equation negative three 𝑥 plus five 𝑦 equals eight. So, B would be our answer.