Rearranging Linear Equations

We have now come to the end of week 9 and the end of our unit on equations of linear relations. This means this will be my final blog post of Math 10. There are many different forms of linear equations that can be used. First there is slope intercept form, y=mx+b. This form shows the slope (m) and the y-intercept (b). Then there is slope intercept form, m (x-x) = y-y. This form shows the slope (m) and a point on the graph. Then there is general form, Ax+By+C=0. This form is the most useful for computer programming and is just a simpler form. However it doesn’t give you a lot of information. We learned about all the different types of equation formats and in this blog post I will be going through two examples of how to convert different formats. I believe that this is quite important as rearranging equations is something that is used in all types of math and algebra very often.

In the first example we are going to be converting an equation from general form to slope-intercept form.

Here is the example question we are going to be working with:

Step 1 – rearranging the equation: first off we need to rearrange the equation to isolate y (get y by itself). In order to do this we need to move 3x and -5 to the other side of the equation. Since they are being moved to the other side of the equal sign their signs will be made opposite. 3x will become -3x and -5 will become 5.

Step 2 – fully isolating y: now we need to get y completely on its own. For this to happen we are going to need to divide all the terms in the equation by 2 so that y will lose its coefficient (or have a coefficient of 1). Some of the terms will not divide evenly so they can be simply left as fractions.

We have now got the equation into slope-intercept form in two simple steps. We can now identify the slope and the y-intercept.

 

Example 2:

We are now going to complete another example except this time we are going to convert an equation in point-slope form to general form. This question will look a little different, instead of going directly from an equation to a different format we will be looking at some words and making use of them.

Here is the example question we are going to be working with:

Write the equation in general form: with slope and an x-intercept of -6.

Step 1 – interpreting the question: so first of all we know that the slope is . And we also know that the x-intercept is -6. When a line intercepts the x-axis y is always equal to 0. This means that we know this line will pass through the point (-6,0). Now since we are given the slope and a point on the line we are able to write the equation in point-slope form. I am aware that the question is not asking for point-slope form, however we can later convert it to general form from point-slope form.

Step 2 – writing the equation in point-slope form: the formula for point-slope form is m(x-x) = y-y. This means that we will write our slope in front of the brackets and x minus our x-coordinate is equal to y minus our y-coordinate. And another thing to remember is that when writing in point slope form you will need to write each coordinate with the opposite sign. For example in this case I will write -6 as +6.

Step 3 – expanding the equation: next we need to simply remove the brackets and multiply by everything in the brackets. The newly formed equation will look like this:

Step 4 – getting rid of the fractions: after we have expanded the equation we will need to remove the fractions since one of the main rules of general form is no fractions allowed. In order to do this we will need to multiply all the terms by a common denominator which in this case is 7.

Note: as you can see we have also now gotten rid of the 0.

Step 5 – rearranging the equation: we are going to need to get all the terms on one side of the equation so that they can equal to 0. In this case we are going to want to move 7y over to the left side since the coefficient of x is positive. If this number was negative which it cannot be in general form we would have to move the terms on the right side to the left. Once we move 7y to the other side of the equation it will become -7y. And finally after that we will make the equation equal to 0. The equation is now in general form.

 

Summary:

We have now completed two different examples of rearranging linear equations into different forms. Once again, I believe that this is a very important skill to master as rearranging equations is a very common thing in many parts of math. Because of this it is crucial to be comfortable with doing it and to understand the topic to a good extent.