Write in the form 𝑦 equals 𝑚𝑥 plus 𝑐 the equation of a line through negative one, negative one that is parallel to the line negative six 𝑥 minus 𝑦 plus four equals zero.
So we’ve been asked to find the equation of a line in slope-intercept form: 𝑦 equals 𝑚𝑥 plus 𝘤. We’ve also been given two pieces of information about this line, which are the coordinates of a point on this line negative one, negative one. We’re also told the key piece of information that this line is parallel to the line, whose equation is negative six 𝑥 minus 𝑦 plus four equals zero. So in order to answer this question, we need to determine the values of 𝑚 and 𝑐 for the line that we’re interested in.
Let’s begin by thinking about 𝑚, the slope of the line. We’re told that this line is parallel to the line, whose equation we’ve been given. And so we need to remember the key fact that if two lines are parallel, then their slopes are the same. This means we can determine the value of 𝑚 for our line by looking at the slope of the other line. Looking at the equation of the second line, it’s not quite in the right format for us to be able to determine its slope. We need to rearrange it into slope-intercept form.
This only requires one step of working out. I need to add 𝑦 to both sides of the equation. And doing so, I have that 𝑦 is equal to negative six 𝑥 plus four. Now this line is in slope-intercept form. And if I compare it to 𝑦 equals 𝑚𝑥 plus 𝘤, I can see that the slope of this line is negative six; it’s this value here — the coefficient of 𝑥. Remember the line we’re interested in is parallel to this line, so it has the same slope. This means that our line has the equation 𝑦 equals negative six 𝑥 plus 𝘤, for a value of 𝘤 that we now need to calculate.
Remember that the point with coordinates negative one, negative one lies on our line. Phrased another way, this means that these values of 𝑥 and 𝑦, which here are both negative one, satisfy the equation of our line. So I can substitute these values into the equation in order to determine 𝘤. So substituting negative one for both 𝑥 and 𝑦, we now have negative one is equal to negative six multiplied by negative one plus 𝑐, and this is an equation that I can solve.
Negative six multiplied by negative one is positive six, so I have negative one is equal to six plus 𝑐. I now need to subtract six from both sides of the equation. And then doing so, I have that negative seven is equal to 𝑐. So we found the value of 𝘤.
The final step is I need to substitute this value of 𝘤 into the equation of the line. So we have our answer to the problem. The equation of this line in slope-intercept form is 𝑦 is equal to negative six 𝑥 minus seven. Remember the key fact that we’ve used in this question was that if two lines are parallel, then their slopes are equal.