Write an Equation: a Standard Approach
If you are on this page, you probably have an understanding of the basic principles and concepts. Only with their help, you will be able to write a standard-form equation. In addition, you can write the equation using a slope-intercept format.
You can also convert the equation from a slope-intercept format into a standard one without any efforts.
Here is a list of the core concepts that we will be using in the article to provide and explain equations. Make sure you know all of them:
- Graphing lines and coordinate plane;
- Y- and slope-intercept;
- Point- and slope-intercept forms;
- Parallel and perpendicular lines equations;
- Integers and variables.
Here is a standard example of an equation form: Ax + By = C, where:
- A, B, C are no-common factor integers (except 1);
- A is non-negative;
- x,y are variables.
The same equation can be written in a slope-intercept form: y=mx +b
How to Convert the Slope-Intercept Form into a Standard One
If you want to change the slope-intercept equation into a standard one, you’ll need to express the y-intercept coordinate and the slope as a two-integers quotient, which is also called the rational-number form. The slope formula is defined as they change divided by the x change.
The slope-intercept equation y= MX + b can be written as y= (y change/ x change) x + b num/ b den. The y-intercept ordinate usually follows the same pattern, so be in the above equation can be changed into two integers quotient, which is called b den and b num.
If you look at a standard formula linear equation, you can see that x and y-intercepts the appropriate line, for example, 4x + 3y = 24.
If we take x value as 0, the equation will look as follows: 3y = 24. Now we can easily state that y value is 8, so the y-intercept equals (8,0). Similarly, if the value of y will be 0, the equation will be 4x = 24 and the x-intercept will equal (0,6).
After multiplying sides of the y = mx + b equation by the multiple of b den and x change, the resulting equation will have no fractions and it will look as follows:
Dy = Ex + F, where D,E,F are integers.
If you want to transform the equation into a standard one, you will need to have x coefficient as non-negative. That is why you’ll need to add -Ex to equation sides:
-Ex + Dy = F. In case sides of the equation are negative, they can be x by -1.
Eventually, we’ll get a standard equation:
Ax + By = C, where A is not negative and where A, B, C are integers. Three of them won’t have any alike factors apart from 1.
Equations, Transformed into a Standard Form
Let us start with a complex example:
y= (5 / 7)x + 7/4
On the first stage let us use a multiple 4 and 6 that equals 12 to multiply sides of the equation.
It will look as follows:
12y = 12( ( 5/6)x + 7/4 )
12y = 10x + 21
Then we need to remove fractions and place x on the left side:
-10 x + 12 y = 21
Considering that x has a negative coefficient, it is necessary to multiply sides by -1.
10x -12y = -21
Here is another example of how to change slope-intercept format into a standard one:
Y= 1/3 x + 5/6
The least multiple for 3 and 6 is 6, so you need to multiply the sides by 6 in order to remove fractions.
6y = 6 ((1/3 x) + 5/6)
6y = 2x +5
Add – to both of the sides
-2x + 6y = 5
To make x coefficient positive you need to multiple sides of the equation by -1. This will help you to get a standard form:
-1 (-2x + 6y) = -1×5
2x – 6y = -5
A simple example of slope-intercept format looks as follows:
y = -3x + 6
In the above example, there is no need to remove fractions. Simply move x to the left to get a standard form:
3x + y = 6
Why You May Need Converting Equations to a Standard Form
In the above examples, we have learned different ways of converting slope-intercept equations into standard ones. But you may still wonder why there may be a need in such transformations.
Here are only some of the reasons:
- Standard equations may be written using vertical lines, while slope-intercept ones can’t be written in the same format;
- The standard form allows using this technique to solve linear equations;
- It becomes much easier to find parallels and perpendiculars with a standard equation.
You should remember that vertical lines are not applied for slope-intercept equations, because it has an undefined slope. However, we can still take points (4,7) for the vertical line. For example, 1x + 0y =4 and you can simply write x = 4.
Keep in mind that a horizontal line at (4,7) has the slope-intercept form y = 0x + 7 that has a standard equation of 0x + 1y = 7.
This explains why it is necessary to have a non-negative x coefficient. For horizontal lines, x coefficient should equal 0.
Problem of a Parallel Line
Let us consider an example, where it is necessary to find the equation for a line that is parallel to 3x + 4y = 17. This line has points (2,8). It is necessary to find this line’s slope and to use it together with the points of the point-slope. Let’s have a look at a standard linear equation:
Ax + By = C
Start with moving Ax to the right:
By = – Ax +C
If we divide all of the sides by B (assuming that it doesn’t equal 0), we will get a slope-intercept equation:
y= (-A /B) x + C/B
You can see that the slope is in the form –A/B. This means that any parallel line to this one should have the same slope.
As long as A and B are not changed, any line of a standard form of Ax + By =H will go the same as Ax + By=C.
Now let us return to the initial problem: to make an equation for a line, which is parallel to 3x + 5y = 17.
This line contains points (2,8). Here is how the result will look like:
3x + 5y= H
This means that the H value must be found. We know that point (2,8) makes the equation fair, so we will conclude that
3×2 + 5×8 = H and that H will equal 46.
Now, when we know that H=46, we can transform our initial equation:
3x + 5y = 46.
When we have a standard linear equation and need to find the equation of a parallel line by using indicated points, we know that the result will look similar to the initial equation. The only difference will be the value because the resulting equation will have a different one.
In order to find the resulting value, we will need to substitute the equation’s point.
For example, 2x + – 6y = -18 with the point (4,-5)
The answer to this equation will be 2x + – 6y = H, where we’ll substitute x and y. The equation will look as follows:
2×4 + – 6(-5) = H, which means that H = 38.
Having the value H, we can find a solution:
2x + – 6x = 38.
Based on the examples above, it is evident that it’s possible to solve problems of parallel and perpendicular lines by making a few steps. We don’t need to know either original slope nor to use the point-slope format.
Standard horizontal equation is 0x + 1y = C.
Parallel line will be 0x +1y =D.
So the rule will be applied for finding parallel lines. To find out the new one you will simply need to find a new D value (substituting external points).
If you reverse values of A and B, and one of the signs (for example, 0x + 1y=C into 1x +0y= D), you’ll get a vertical line based on the method of distinguishing perpendicular lines.