A line 𝐿 has a slope of three and passes through the point three, four. Work out the equation of the line, giving your answer in the form 𝑦 equals 𝑚𝑥 plus 𝑐.
So we’ve been asked to give the equation of the line 𝐿 in the form 𝑦 equals 𝑚𝑥 plus 𝑐. Now, this is known as the slope-intercept form, where 𝑚 represents the slope of the line and 𝑐 represents the 𝑦-intercept.
We’ve been told in the question that our line has a slope of three. So straightaway, we can fill in one piece of information. And we know that the equation of line 𝐿 is of the form 𝑦 equals three 𝑥 plus 𝑐. We just need to find the value of 𝑐.
Now, there are two methods that we can use to work out the value of 𝑐 both, using the fact that the line passes through the point with coordinates three, four. This means that when 𝑥 is equal to three, 𝑦 is equal to four. So this pair of values need to satisfy the equation of the line. We can substitute the value four for 𝑦 and the value three for 𝑥. And it gives four is equal to three multiplied by three plus 𝑐.
We can then solve this equation for 𝑐. Three multiplied by three is nine. So we have four is equal to nine plus 𝑐. To find the value of 𝑐, we need to subtract nine from each side of the equation. And it gives negative five is equal to 𝑐. So that’s the first method for calculating 𝑐.
And the second is to use what’s known as the point-slope form of the equation of a straight line: 𝑦 minus 𝑦 one is equal to 𝑚 𝑥 minus 𝑥 one. Here, 𝑚 represents the slope of the line as it has before. And 𝑥 one, 𝑦 one give the coordinates of a point that we know lies on the line. So in this case, that’s the point three, four.
We can substitute the slope of three and the values of 𝑥 one and 𝑦 one as three and four into this form of the equation of our line. And it gives 𝑦 minus four is equal to three multiplied by 𝑥 minus three. We just need to rearrange this equation into the required form. Expanding the brackets on the right gives three 𝑥 minus nine. So we have 𝑦 minus four is equal to three 𝑥 minus nine.
The final step in rearranging is to add four to each side of the equation. And it gives 𝑦 is equal to three 𝑥 minus five. This gives the value of 𝑐 as negative five, which is the same as the value we found when we used the first method. You can use either of these two methods, depending on which you feel most comfortable with.
The equation of line 𝐿 in the form 𝑦 equals 𝑚𝑥 plus 𝑐 is 𝑦 equals three 𝑥 minus five.