Consider the line with the equation 𝑦 equals negative two 𝑥 plus five. Circle the equation of a line which is parallel to this line. The options are 𝑦 equals a half 𝑥 minus two, 𝑦 equals negative a half 𝑥, 𝑦 equals two 𝑥 minus two, or 𝑦 equals negative two 𝑥 plus 11.
First, we need to remember what it means for two lines to be parallel. In general terms, two lines are parallel if they are always exactly the same distance apart. They never get closer together or further away from each other.
If we’re given the equations of two lines, then these lines are parallel if they have the same gradient or steepness. We can work out the gradient of any straight line by recalling that the general form of the equation of a straight line is 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 represents the gradient of the line. 𝑐 represents the 𝑦-intercept, but this isn’t actually relevant for this question.
Looking at the line whose equation we’ve been given, we can see that it is in the general form. And the number in front of the 𝑥 — that’s the coefficient of 𝑥 — is negative two, which means that the gradient of this line is negative two.
To work out which of these four lines is parallel to our line, we need to work out each of their gradients. Now fortunately, they’re all in the form 𝑦 equals 𝑚𝑥 plus 𝑐. We don’t need to do any rearranging. So this is a relatively straightforward task.
For the first line, the coefficient of 𝑥 is one-half. So this is its gradient. One-half is not the same as negative two. So this line isn’t parallel to our line.
For the second line, the coefficient of 𝑥 is negative a half. So this is the gradient of this line. We do need to make sure we pay close attention to the signs. We can also rule out this line.
For the third equation, the coefficient of 𝑥 is two. So the gradient of this line is two. Again, remember, we need to pay close attention to the signs. Two is not the same as negative two. So this line is not parallel to our line.
For the final line, the coefficient of 𝑥 is negative two. So this line also has a gradient of negative two. Our answer is that the line 𝑦 equals negative two 𝑥 plus 11 is parallel to the line 𝑦 equals negative two 𝑥 plus five.
Actually, the first line we were given does have a special property in relation to the line 𝑦 equals negative two 𝑥 plus five. It’s perpendicular to this line, meaning the two lines meet at right angles. We know this because the product of their gradients, a half and negative two, is negative one. And if this is the case, then two lines are perpendicular.
However, in this question, we were asked for the line which is parallel to our line, not perpendicular. So our answer is 𝑦 equals negative two 𝑥 plus 11.