Video Transcript
Determine the equation of the line
which cuts the 𝑥-axis at four and the 𝑦-axis at seven.
Although it isn’t entirely
necessary, we may find it helpful to sketch this straight line. It cuts the 𝑥-axis at four, and it
cuts the 𝑦-axis at seven. So it looks something like
this. We can see that this line slopes
downwards from left to right, and so it has a negative slope. This will give us one way of
checking whether our answer, when we reach it, seems reasonable. To answer this question, we’ll
begin by using the slope–intercept form of the equation of a straight line: 𝑦
equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope of the line and 𝑏 is its
𝑦-intercept. We already know the value of 𝑏
because it was given in the question. We were told the line cuts the
𝑦-axis at seven. So our line is 𝑦 equals 𝑚𝑥 plus
seven. And we need to calculate its
slope.
The slope of a straight line
passing through the two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two is
𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. Using the coordinates of the 𝑥-
and 𝑦-intercepts then, which are four, zero and zero, seven, we can calculate the
slope of this line. It’s zero minus seven over four
minus zero, which simplifies to negative seven over four. This is a negative value, so we can
be reassured that what we’ve done so far is correct. The equation of this line, then, is
𝑦 equals negative seven over four 𝑥 plus seven.
Now we could leave our answer in
this form, but it looks a little unfriendly because we have a quotient. We can find an equivalent form of
this equation by rearranging to the general form. We’ll begin by multiplying every
term in the equation by four, giving four 𝑦 equals negative seven 𝑥 plus 28. Next, we’ll group all of the terms
on the same side of the equation, choosing the left-hand side so that the
coefficients of both 𝑥 and 𝑦 are positive. So adding seven 𝑥 and subtracting
28 from each side, we have seven 𝑥 plus four 𝑦 minus 28 is equal to zero. This is the equation of this
straight line in its most general form. That’s 𝑎𝑥 plus 𝑏𝑦 plus 𝑐
equals zero, where 𝑎, 𝑏, and 𝑐 are real constants.
