Video Transcript
In the 𝑥𝑦-plane shown, line 𝑙 one is parallel to line 𝑙 two. What is the value of 𝑘?
Based on the given information, we know that line 𝑙 one and 𝑙 two are parallel. We also know that parallel lines have the same slope. The slope of the line is the changes in 𝑦 over the changes in 𝑥. If we want to calculate the slope of line one, we notice that it is going up three. The changes in 𝑦 are positive three. And that it goes to the right four. So, the changes in 𝑥 are positive four. And the slope of line one is three-fourths. It also means the slope of line two is three-fourths.
If we start at the point zero, negative six and go up three and right four, we would be at the point four, negative three. We’ve added four to the 𝑥-axis and added three to the 𝑦-axis. If we repeat this process and go up three again, we’ll be at the coordinate zero along the 𝑦-axis. And we move four to the right again. Four plus four equals eight. And this means we are at the point eight, zero. 𝑘 must be equal to eight.
Sometimes we write the formula for slope like this, 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. Instead of sketching the slope on the graph, we could’ve used the two points to find the slope of line one. We would label the coordinates, 𝑥 one, 𝑦 one, 𝑥 two, 𝑦 two. Our numerator would then be zero minus three. And our denominator would be negative four minus zero. This gives us negative three over negative four. And we simplify that to three-fourths.
And then, we would repeat this process for line two. We would label one of the points 𝑥 one, 𝑦 one and the other point 𝑥 two, 𝑦 two on the line 𝑙 two. The numerator would be zero minus negative six. And the denominator would be 𝑘 minus zero. Based on the information from line one, we know that the slope must be three-fourths. So, we plug in three-fourths as the slope of line two. Zero minus negative six is positive six. And 𝑘 minus zero is 𝑘. If three-fourths is equal to six over 𝑘, what is 𝑘 equal to? We know that three times two is six. And that means 𝑘 has to be equal to four times two, which confirms that 𝑘 equals eight.
