Video: Graphing Linear Equations by Changing Their Form

Consider the equation 3𝑦 = (6π‘₯ + 3)/(2). Rearrange the equation into the form 𝑦 = π‘šπ‘₯ + 𝑐. What are the slope and 𝑦-intercept of the equation? Use the slope and intercept to identify the correct graph of the equation.

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Video Transcript

Consider the equation three 𝑦 equals six π‘₯ plus three over two. Rearrange the equation into the form 𝑦 equals π‘šπ‘₯ plus 𝑐. What are the slope and 𝑦-intercept of the equation? Use the slope and intercept to identify the correct graph of the equation.

Let’s begin this question by having a look at the first part. Here, we’re asked to rearrange our equation into the form 𝑦 equals π‘šπ‘₯ plus 𝑐. We can remember that this form, 𝑦 equals π‘šπ‘₯ plus 𝑐 or often 𝑦 equals π‘šπ‘₯ plus 𝑏, is the general form which allows us to identify key parts of a graph. We can see that our equation has a 𝑦 and an π‘₯ in the same way that the general form has a 𝑦 and an π‘₯. The difference here is that we have a three 𝑦 instead of just 𝑦. So we’ll need to divide both sides of our equation by three.

This means that, on the right-hand side, we’ll be dividing by two multiplied by three. And as two times three is six, we have 𝑦 equals six π‘₯ plus three over six. In order to simplify this fraction, we can consider the right-hand side as six π‘₯ over six plus three over six. This is equivalent to 𝑦 equals π‘₯ plus a half. And this means we have rearranged the equation into the form 𝑦 equals π‘šπ‘₯ plus 𝑐. For the second part, we can recall that when we have 𝑦 equals π‘šπ‘₯ plus 𝑐, then the coefficient of π‘₯, the letter π‘š, indicates the slope or gradient of an equation. The constant term 𝑐 represents the 𝑦-intercept.

So when we take our equation in this form, that’s 𝑦 equals π‘₯ plus a half, the slope will be represented by the coefficient of π‘₯, which in this case would be one. The 𝑦-intercept will be positive one-half, which we can write as a half. And so we have answered the second question. We can now take a look at the graph options for the third part of the question. We’ve established that the 𝑦-intercept of our graph will be at a half. We can see on our first graph that the 𝑦-intercept is a half since the graph crosses through half on the 𝑦-axis. So this may be a possible answer.

On the second graph, it crosses the 𝑦-axis at negative a half. And so we can rule out the second graph. The third graph has a 𝑦-intercept of one, so this does not fit. And the fourth has a 𝑦-intercept at negative one. The final graph has a 𝑦-intercept of negative a half. So this wouldn’t work either. This means that we have one answer left, but it is worth checking if the slope is equal to one.

We can recall that, to find the slope of a line between two coordinates π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two, we calculate 𝑦 two minus 𝑦 one over π‘₯ two minus π‘₯ one. We can select any two coordinates that lie on the line. Here we have zero, a half and one, three over two. It doesn’t matter which one we designate as π‘₯ one, 𝑦 one and which we designate as π‘₯ two, 𝑦 two. So to find the slope, we have three-halves subtract a half over one minus zero. And since three-halves subtract a half is one and one subtract zero is one, we have one over one, which means that the slope is indeed equal to one. And so our answer is that it is the first graph which represents the equation three 𝑦 equals six π‘₯ plus three over two.

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