Which of the following is an equation of line 𝑙 in the 𝑥𝑦-plane? Is it A) 𝑦 equals 𝑥 plus two, B) 𝑦 equals two 𝑥, C) 𝑥 equals two, or D) 𝑦 equals two?
Any diagonal line in the 𝑥𝑦-plane will have equation 𝑦 equals 𝑚𝑥 plus 𝑏, where the value of 𝑚 is the slope or gradient of the line and the value of 𝑏 is the 𝑦-intercept. This means that we can immediately rule out option C and option D. The equation 𝑥 equals two will be a vertical line through the point two on the 𝑥-axis. The equation 𝑦 equals two would be a horizontal line through the point two on the 𝑦-axis. Let’s now consider the line 𝑙 that has been drawn.
The line intersects the 𝑦-axis at the point two. This means that the 𝑦-intercept or 𝑏 will be equal to two. We can therefore say that our equation will be of the form 𝑦 equals something 𝑥 plus two. This rules out option B, which suggests that option A is the correct answer. We can check this by working out the gradient of line 𝑙. The gradient or slope of a line could be calculated by dividing the change in the 𝑦-coordinates by the change in the 𝑥-coordinates. This is often known as the rise over the run. In order to do this, we create a right angle triangle, as shown.
The change in the 𝑦-coordinates here is two as two minus zero equals two. The change in the 𝑥-coordinates is also two, as zero minus negative two is equal to two. This means that our slope or gradient is equal to two divided by two. This equals one. So the value of 𝑚, the coefficient of 𝑥, is one. The equation of line 𝑙 is 𝑦 equals one 𝑥 plus two. This can also be written as 𝑦 equals 𝑥 plus two.
The correct answer from the four options was option A. 𝑦 equals 𝑥 plus two is an equation of line 𝑙. It is worth remembering that if our line slopes up from left to right, it will have a positive slope or gradient. And if it slopes down from left to right, it will have a negative slope or gradient. In this case, as line 𝑙 sloped upwards, the gradient had to be positive.