Given that the slope of a straight line passing through points nine, negative seven and negative three, 𝑘 is negative five twelfths, find the value of 𝑘.
We have two points and a slope. The variable 𝑚 usually represents slope, which is the changes in 𝑦 over the changes in 𝑥. Given our two points, we would say 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. Let’s label our two points. Nine, negative seven turns into 𝑥 one, 𝑦 one. Negative three, 𝑘 is 𝑥 two, 𝑦 two.
We’ll plug those points into our formula 𝑘 minus negative seven over negative three minus nine. And remember that we already know the slope. Negative five over 12 is equal to 𝑘 minus negative seven over negative three minus nine. We can do a little bit of simplification. 𝑘 minus negative seven is equal to 𝑘 plus seven. Negative three minus nine equals negative 12.
Our new statement says negative five over 12 is equal to 𝑘 plus seven over
12 [negative 12]. We want to isolate 𝑘. To do that, I want to multiply the right side by negative 12 over one. If we multiply the right side by negative 12 over one, we need to multiply the left side by negative 12 over one.
On the right side, the negative 12s cancel each other out and we’re left with 𝑘 plus seven. On the left side, the 12s cancel out. But we notice that we’re multiplying a negative by a negative. And that means we’re left with positive five. Five equals 𝑘 plus seven.
To isolate 𝑘, we subtract seven from both sides of the equation. The sevens cancel out. 𝑘 is equal to five minus seven, which is negative two.
𝑘 equals negative two.