# Video: Finding the Equations of Parallel Lines

Practise finding the equation of parallel lines using the fact that their slopes are equal.

11:03

### Video Transcript

In this video, we’re going to look at the relationship that exists between the equations of parallel lines and then see how to find the equation of some parallel lines.

Firstly, I’ll remind of what parallel lines are. Two lines are parallel if they will never meet, no matter how far they’re extended in either direction. They always remain exactly the same distance apart. In the diagram here, I’ve drawn a pair of parallel lines and labeled them as 𝑦 equals 𝑚 one 𝑥 plus 𝑐 one and 𝑦 equals 𝑚 two 𝑥 plus 𝑐 two, both in slope-intercept form. The key fact about parallel lines is that their slopes are equal. In the context of their equations then, 𝑚 one is equal to me two. This means that if we know the equation of one of the lines or can calculate its slope, then we’re halfway to finding the equation of the other line, as the slope will be the same. Within this video, we’ll look at finding the equations of parallel lines within various different contexts.

Write, in the form 𝑦 equals 𝑚𝑥 plus 𝑐, the equation of the line that is parallel to the line negative four 𝑥 plus seven 𝑦 minus four equals zero and that intercepts the 𝑦-axis at one.

So we’re told the format that our answer should take. It needs to be in slope-intercept form. Remember here, the 𝑚 represents the slope and 𝑐 represents the 𝑦-intercept. Looking at the question, we can see that we’re given one of these values directly. We’re told that this line intercepts the 𝑦-axis at one. This means then that the value of 𝑐 is equal to one. So the equation of a line is 𝑦 equals 𝑚𝑥 plus one. And we’re already halfway to our answer. In order to find the value of 𝑚, the slope, we need to use the fact that this line is parallel to the line negative four 𝑥 plus seven 𝑦 minus four equals zero. If they’re parallel, then they have the same slope. So what I’m going to do is, I’m going to rearrange the equation of this line to get it into slope-intercept form.

So I begin with negative four 𝑥 plus seven 𝑦 minus four is equal to zero. I’m going to add both, four 𝑥 and four, to both sides of this equation. This gives me seven 𝑦 is equal to four 𝑥 plus four. Next, I’m going to divide both sides of this equation by seven. This gives me 𝑦 is equal to four-sevenths of 𝑥 plus four-sevenths. Now if I compare this to the slope-intercept form of a line, I can see that the slope of this line is four-sevenths. It’s this value here. Now the line that I’m looking for is parallel to this line. Therefore, it must have the same slope. So it’s slope is also four-sevenths. The final step in answering this question is to substitute this value of 𝑚 into the equation of my line.

So I have that the equation of this line, in the requested form, is 𝑦 equals four-sevenths 𝑥 plus one. Find, in slope-intercept form, the equation of the line parallel to 𝑦 equals negative eight over three 𝑥 plus three that passes through point 𝐴, negative three, two.

So we’re asked to find the equation of this line in slope-intercept form, that is 𝑦 equals 𝑚𝑥 plus 𝑐. What we need to do, is determine the values of 𝑚, slope, and 𝑐, the 𝑦-intercept of the line. We’re told in the question that the line is parallel to the line 𝑦 equals negative eight over three 𝑥 plus three, which means it must have the same slope. The slope of this line is negative eight over three. And therefore, this is the value of 𝑚 in the line that we’re looking for. So we have 𝑦 is equal to negative eight over three 𝑥 plus 𝑐.

Now I need to determine the value of 𝑐. We’re told in the question that the line passes through the point negative three, two. Put another way, this means that within this equation, when 𝑥 is equal to negative three, 𝑦 must be equal to two. We can substitute these values of 𝑥 and 𝑦 into our equation in order to find the value of 𝑐. So by substituting the value of negative three for 𝑥 and the value of two for 𝑦, I now have the equation two is equal to negative eight over three multiplied by negative three plus 𝑐. This simplifies to two equals eight plus 𝑐. To solve this equation for 𝑐, I need to subtract eight from both sides. And in doing so, I get that the value of 𝑐 is negative six. The final step then is I need to substitute this value of 𝑐 into my equation for the line.

So we have the equation of the line is 𝑦 is equal to negative eight over three 𝑥 minus six.

In the figure below, 𝐿 one is parallel to 𝐿 two and 𝐴𝐵 equals eight length units. If the equation of 𝐿 one is 𝑦 equals four-fifths of 𝑥 plus four, find the equation of 𝐿 two.

So within this question, 𝐿 one and 𝐿 two are the labels given to these two straight lines. This piece of notation here means that line one is parallel to line two. So we’re given the equation of line one and we’re asked to find the equation of line two. We’re also given the key piece of information that the two lines are parallel. As the lines are parallel, this means that they have the same slope. So I’m going to find the equation of this line in slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. We can see from the equation of 𝐿 one that its slope is four-fifths. And as the lines are parallel, the slope of line two is the same. Therefore, the value of 𝑚 must be four-fifths, and we have the beginnings of our equation for line two.

Next, we need to work out the value of 𝑐, the 𝑦-intercept of line two. The other piece of information we’re given in the question is that 𝐴𝐵 is eight length units. So the length of this vertical section here is eight length units. This means that the difference between the 𝑦-intercepts of line one and line two must be eight. Looking at the equation of line one, we can see that its 𝑦-intercept is equal to four. So to find the 𝑦-intercept of the second line, we just need to subtract eight from four. This tells us then that the 𝑦-intercept of line two is negative four. Finally, I just need to substitute this value for 𝑐 into the equation of the line.

I have then that the equation of line two is 𝑦 equals four-fifths of 𝑥 minus four.

What is the value of 𝑏 if the lines negative two 𝑥 plus 𝑏𝑦 plus six equals zero and negative 𝑥 minus four 𝑦 minus three equals zero are parallel?

So we’re given the equations of two lines, and we’re told the key piece of information that they are parallel. Remember, this means that the slopes of the two lines are equal. I’d like to find the slopes of the two lines. It isn’t immediately obvious from the format that they’re currently in. So I’m going to rearrange the equation of each line into slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. So starting with the line negative 𝑥 minus four 𝑦 minus three equals zero first of all. I’m going to begin by adding four 𝑦 to both sides of the equation. This gives me four 𝑦 equals negative 𝑥 minus three. I then need to divide both sides of the equation by four. This gives me 𝑦 equals negative 𝑥 over four minus three-quarters. Or you could think of it as minus a quarter 𝑥 minus three-quarters. Comparing this with the slope-intercept form of the equation of the line, I can see that the slope of this line is negative a quarter.

Now I want to find the slope of the second line. And it will be in terms of this unknown letter 𝑏. So I have negative two 𝑥 plus 𝑏𝑦 plus six is equal to zero. So I’m gonna group the terms that don’t involve 𝑦 on the right-hand side of the equation, which means I’m going to be adding two 𝑥 and subtracting six. This gives me the equation 𝑏𝑦 equals two 𝑥 minus six. Next, I need to divide both sides of the equation by this unknown value 𝑏. So I have 𝑦 is equal to two over 𝑏 𝑥 minus six over 𝑏. Now comparing this with the slope-intercept form of the equation of a line, I can see the slope of this line is two over 𝑏. So I have the slope of each line, one exactly and one in terms of this unknown letter 𝑏.

Now remember, these lines are parallel, and therefore their slopes are equal to each other. This means that I can form an equation to solve in order to find the value of 𝑏. So I have negative a quarter is equal to two over 𝑏. In order to solve this equation, I’d like to multiply by 𝑏 as it’s currently in the denominator of our fraction. And I’d also like to multiply by four. So cross multiplying gives me negative one multiplied by 𝑏 is equal to two multiplied by four. Or equivalently, negative 𝑏 is equal to eight. Finally, I just need to divide both sides of the equation by negative one, and this gives me the value of 𝑏. And therefore, my answer to the problem: 𝑏 is equal to negative eight.

In summary then, we’ve seen that a key property of parallel lines is that their slopes are equal. If the lines are represented in slope-intercept form, 𝑦 equals 𝑚 one 𝑥 plus 𝑐 one and 𝑦 equals 𝑚 two 𝑥 plus 𝑐 two, then this means that 𝑚 one and 𝑚 two must be equal to each other. In this video, we’ve seen how to use this key property along with our other methods for finding the equation of a straight line in order to find the equation of lines that are parallel to each other.