Basic Definition about Equations
Equations are expressions with equal signs ("="). They represent two things or expressions that are equal.
For example:
5 + 4 = 9
5 + 4 = 3 + 6
Another instance for an equation:
5x + 6 = 10
Means: 5x, 6, and 10 are terms.
5x + 6 is an expression.
Linear Equation
Linear Equation is an equation that has a straight line.
Forms of linear equation are varied, as follows:
- Slope-intercept form: y= mx + b (m = slope and b = y-intercept)
- Point-slope form: y - y1= m(x - x1) (m = slope and (x1, y1) = point in the line)
- Standard form: Ax + By = C
- Horizontal line form: y= b (horizontal line with b as the y-intercept)
- Vertical line form: x = a (vertical line with a as the x-intercept)
To find slope, formula we can use:
| m = | y2-y1 |
| x2-x1 |
(if two points on a line are known.)
Or, use Rise/Run
method if we know the graph.
Rise is for the difference between two y-points. If the Rise's direction is downward, the Rise is a negative value, and if the Rise's direction is upward, the Rise is a positive value.
Run is for the difference between two x-points. If the Run's direction is leftward, the Run is a negative value, and if the Run's direction is rightward, the Run is a positive value.
Always consider using a right triangle concept between two points.
Example:
To find the slope for this example above, we can use two ways:
1. Use formula
| m = | Rise |
| Run |
Rise = y2-y1 = 10 - 0 = 10
Run = y2-y1 = 0 - (-5) = 0 + 5 = 5
| m = | Rise |
| Run |
| = | 10 |
| 5 |
2. The second method is to use formulas:
| m = | y2-y1 |
| x2-x1 |
Two points in the graph are Point 1 (0, 10) and Point 2 (-5, 0), so by using the slope formula above, we can calculate the slope, as follows:
| m = | y2-y1 |
| x2-x1 |
| = | 0-10 |
| -5-0 |
| = | -10 |
| -5 |
= 2
By using two different methods above, we can see that we obtain the same results. To find the equation, we use a formula:
y = mx + b.
b represents y-intercept.
It means x = 0. From the graph above, Point 1 (0, 10) has the y-intercept, which is 10. We already know that m = 2 and b = 10. Input those two values into the formula above:
y = mx + b
Final Equation (Slope-intercept form):
y = 2x + 10
Change the forms of equations
From the example above, y = 2x + 10 is a slope-intercept form. We can convert the form into other forms:
A. Standard Form (Ax + By = C)
y = 2x + 10
-2x + y = 10 ----------- Subtract both sides by 2x
OR
2x - y = -10 ----------- Multiply all values by -1
The final result of Standard Form should be:
-2x + y = 10
OR
2x - y = -10
B. Point-Slope Form (y - y1= m(x - x1))
For this form, we must choose one point of the equation. In this example, choose one out of the two points above (Point 1 (0, 10) and Point 2 (-5, 0)).
Here I choose Point 1.
Input the Point 1 (0, 10) and the slope (m = 2) into the formula:
y - y1= m(x - x1)
y - 10 = 2(x - 0)
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