A line has a slope of negative three two and passes through the point five, zero. What is the equation of this line?
We have the slope and a point. Our first step will be using the point-slope form: 𝑦 minus 𝑦 one equals 𝑚 times 𝑥 minus 𝑥 one. Our point five, zero is our 𝑥 one, 𝑦 one. And the 𝑚 value is the slope, negative three-halves. We’ll plug this information in: 𝑦 minus zero equals negative three-halves times 𝑥 minus five. On the left, 𝑦 minus zero equals 𝑦, negative three-halves times 𝑥 equals negative three-halves 𝑥, and then negative three-halves times negative five equals positive 15 over two.
The equation we have now is in slope form. But we want to write the equation for this line in standard form. In standard form, the 𝑥 is positive and none of the coefficients are fractional. Not only that, the whole equation is set equal to zero. We have a negative 𝑥 that needs to be moved to the left side of the equation. So we add three-halves 𝑥 to the right and the left. Negative three-halves 𝑥 plus positive three-halves 𝑥 equals zero. Our new equation says three-halves 𝑥 plus 𝑦 equals 15 over two.
We want the right side to be equal to zero. So we subtract 15 over two from the right and the left. Positive 15 over two minus 15 over two equals zero. And the left side says three-halves 𝑥 plus 𝑦 minus 15 halves equals zero. Well let’s give ourselves a little bit more space. Remember, at the beginning I said that our 𝑥 needed to be positive and there could be no coefficients that are fractional. We can’t have these divided by two pieces.
To get rid of that we’ll multiply the whole equation by two. Two times three-halves 𝑥 equals three 𝑥. Two times 𝑦 equals two 𝑦. Two times negative 15 halves equals negative 15. And two times zero equals zero. The equation for this line is three 𝑥 plus two 𝑦 minus 15 equals zero.