What is the 𝑦-intercept of the line passing through negative two, negative 16 and one, negative four.
We’ve been given two points that lie along the same line. And we’re trying to find the 𝑦-intercept of this line. If the line is given in the form 𝑦 equals 𝑚𝑥 plus 𝑏, the constant value 𝑏 represents the 𝑦-intercept.
In order to take these two points and write an equation for this line, we’ll first need to calculate the slope. If we’re given two points, we can find the slope by taking the 𝑦-coordinate of the second point, subtracting the 𝑦-coordinate of the first point, and writing that as a fraction over the 𝑥-coordinate of the second point minus the 𝑥-coordinate of the first point. More commonly, we just say 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. It doesn’t matter which point we label as point one and which point we label as point two as long as we’re consistent.
So we have negative four minus negative 16 in the numerator and one minus negative two in the denominator. Negative four minus negative 16 is negative four plus 16, which is 12. And one minus negative two is one plus two, which is three. The slope of this line is four. We can take this slope and one of these points and use it to help us solve for the 𝑦-intercept.
If 𝑦 equals 𝑚𝑥 plus 𝑏 and our slope is four and we have the point one negative, four. We can plug in one for the 𝑥-value and negative four for the 𝑦-value. Four times one is four. And so we can say that negative four equals four plus 𝑏. To find what 𝑏 is, we subtract four from both sides of the equation. Negative four minus four is negative eight. And four minus four is zero. Our 𝑏-value is then equal to negative eight. Our equation of this line must be 𝑦 equals four 𝑥 minus eight. And we’re only looking for the 𝑦-intercept, which is negative eight.
We solved this problem finding the slope and then the 𝑦-intercept form of the line. We could also try and solve by graphing. Once you sketch a graph, we’ll add a point at negative two, negative 16 and one, negative four and then connect those points. We can see that this line crosses the 𝑦-axis at negative eight, which confirms the 𝑦-intercept of negative eight. Now, the key here to solve by graphing is that you need to be accurate with your lines and spaces. You would likely want to draw this on graph paper to make sure you were getting accurate points. Otherwise, you can solve using the slope-intercept form.