Which of the following equations represents a line that is parallel to the line with the equation 𝑦 equals two 𝑥 minus seven? A) Two 𝑥 plus 𝑦 equals seven, B) seven 𝑥 plus 𝑦 equals two, C) four 𝑥 plus two 𝑦 equals three, or D) six 𝑥 minus three 𝑦 equals five.
Our equation is 𝑦 equals two 𝑥 minus seven and it’s in the form 𝑦 equals 𝑚𝑥 plus 𝑏. In this form, the 𝑚 equals the slope. And that’s important because parallel lines have slopes that are equal. One thing we could do is identify the slope of all four of these equations.
Starting with A, we have two 𝑥 plus 𝑦 equals seven. We need to rearrange this and put it in slope-intercept form. So we subtract two 𝑥 from both sides, and we get 𝑦 equals negative two 𝑥 plus seven. Remember, we’ve said that the coefficient of 𝑥 — the 𝑚 value — is the slope. The slope of our original equation is two. If we’re looking for a parallel line, we’re looking for another line that has a slope of two. But the slope of A is negative two. And so that won’t work.
What about our equation B, seven 𝑥 plus 𝑦 equals two? We can subtract seven 𝑥 from both sides. And then, we’ll have 𝑦 equals negative seven 𝑥 plus two. The slope is negative seven. So option B won’t work.
Option C, four 𝑥 plus two 𝑦 equals three. We’ll subtract four 𝑥 from both sides and then two 𝑦 equals negative four 𝑥 plus three. However, slope-intercept form has a 𝑦 with a coefficient of one. To get that, we’ll need to divide two 𝑦 divided by two and we’ll have to divide both of the other terms by two as well. Negative four 𝑥 divided by two is negative two 𝑥. And since we’re not really interested in that constant variable, we don’t have to simplify. We can just leave it three over two. Option C has a slope of negative two. Again, we’re looking for a slope of positive two.
So let’s finally check option D: six 𝑥 minus three 𝑦 equals five. Subtract six 𝑥 from both sides and we get negative three 𝑦 equals negative six 𝑥 plus five. Again, we want that coefficient of 𝑦 to be one. So we’ll divide through by negative three. And then, we’ll have 𝑦 equals negative six 𝑥 divided by negative three equals two 𝑥. And we’ll leave five divided by negative three as negative five-thirds. This time we do see a slope of two, which means that the line in option D is parallel to our original equation 𝑦 equals two 𝑥 minus seven.
Solving with this method required us to rearrange four different equations. I want to show you one more method that would only require you to rearrange one of the equations. If we go back to our original equation of 𝑦 equals two 𝑥 minus seven, if we look at our four answer choices, they’re all in the same form: the 𝑥 variable plus the 𝑦 variable equals the constant. We could rewrite our original equation so that it’s in the form the 𝑥 variable plus the 𝑦 variable equals the constant.
To do this, we subtract two 𝑥 from both sides and we get negative two 𝑥 plus 𝑦 equals negative seven. And then, we pay close attention to the coefficients of the 𝑥 and 𝑦 variables. They’re in the ratio of negative two to one. And so, we can look for the other equation that’s also in the ratio negative two to one.
Option A is in the ratio two to one. That won’t work. Option B is in the ratio seven to one. That also won’t work. Option C has coefficients of four to two which is the ratio of two to one. Remember, we’re looking for negative two to one. And then, if we look at six, negative three, at first it might not seem like they’re in this ratio. But if you divided negative three by negative three, you would get one. And if you divided six by negative three, you would get negative two.
Because six, negative three is in the same proportion as negative two to one, we know that the slopes will be the same. Using this method, you wouldn’t have to put all four of the equations into 𝑦-intercept form before you solved.