Write the equation represented by the graph shown. Give your answer in the form 𝑥 plus 𝑦 equals 𝑐.
So we’ve been given a diagram of a coordinate grid and a line whose equation we’re asked to find. We’re going to approach this question by finding both, the slope and the 𝑦-intercept of this line. And then we may need to do some algebraic rearranging of our answer in order to bring it into the requested form.
So let’s look carefully at the diagram, first of all. We can see the coordinates of the points at which the line cuts both, the 𝑥- and 𝑦-axes. In fact, it cuts both axes at a value of five. We’ll be able to use these two points to find both, the slope and the 𝑦-intercept of our line. So we begin with the equation of our line in slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐.
Let’s look at finding the value of 𝑐, first of all. From the diagram, we can see that the line cuts the 𝑦-axis at a value of five. And therefore, 𝑐 is equal to five. We can substitute this value 𝑐 into the equation of our line. So we have 𝑦 is equal to 𝑚𝑥 plus five.
Next, we need to calculate the slope of the line, 𝑚. Given the coordinates of two points on the line, the slope can be calculated as 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one. So let’s look carefully at the two points we know and determine their coordinates. The point where the line crosses the 𝑦-axis has an 𝑥-coordinate of zero, and so the coordinates of this point are zero, five. The point where the line crosses the 𝑥-axis has a 𝑦-coordinate of zero, and so the coordinates of this point are five, zero.
Now we can substitute these values into our calculation of the slope of the line. Remember, it doesn’t matter which point we refer to as 𝑥 one, 𝑦 one and which we refer to as 𝑥 two, 𝑦 two. I’ve chosen purely at random to think of the two points in this order. So the slope then, first of all, 𝑦 two minus 𝑦 one, this is zero minus five. Then I need to divide by 𝑥 two minus 𝑥 one which is five minus zero. This simplifies to negative five over five which is just equal to negative one.
So now we’ve calculated the slope of the line. We can substitute this value of 𝑚 into our equation. So we have 𝑦 is equal to negative 𝑥 plus five. Remember in algebra, when we have a coefficient of one or negative one, we don’t write one 𝑥 or negative one 𝑥; we just write 𝑥 or negative 𝑥.
Now we’ve nearly finished, but if we look back at the question, we see that we were asked for our answer in a specific format, 𝑥 plus 𝑦 is equal to 𝑐. So in order to bring my answer into this format, there’s just one more step. I need to add 𝑥 to both sides of this equation.
And having done so, I now have our answer to the problem: 𝑥 plus 𝑦 is equal to five.