Concept of y-intercept of an equation
To determine the y-intercept of a slope intercept form, we must follow the two requirements: (1) The y-variable must be completely isolated. (2) If case 1 is followed, then the y-intercept is the numerical value in the equation.
Problem 1:
Given equation: y = 6x + 2
Step-by-step explanation: We can see that the "y" variable is completely isolated. Therefore, this equation follows the first requirement. So, the y-intercept is the numerical value in the equation. Clearly, the only numerical value of the provided equation is 2. Therefore, the y-intercept is (0, 2).
Problem 2:
Given equation: 10x + 5y = 30
Step-by-step explanation: We can see that the y-variable is not completely isolated. Therefore, we have to isolate the y-variable using the four operations. To consider the y-variable as isolated, it must not have anything being divided, multiplied, subtracted, or added. In this equation, we can see the y-variable being multiplied to 5. We can also see that 5y is being added to 10x. Therefore, we must subtract 10x on both sides of the equation.
- ⇒ 10x + 5y = 30
- ⇒ 10x - 10x + 5y = 30 - 10x
- ⇒ 5y = 30 - 10x
To remove the coefficient, we must divide both sides by 5.
- ⇒ 5y/5 = (30 - 10x)/5
- ⇒ y = 6 - 2x
Now, since the y-variable is isolated completely, this equation satisfies the first requirement. Clearly, the only numerical value of the equation is 6.
Therefore, the y-intercept is (0, 6).
Problem 3:
Given equation: y - 6 = 2(3x - 4)
Step-by-step explanation: This equation requires some simplifying. Let us first apply the distributive property to simplify the right-hand-side of the equation.
- ⇒ y - 6 = 2(3x - 4)
- ⇒ y - 6 = 6x - 8
We do see two numerical values, but there can only be one y-intercept in any equation when graphed on a coordinate plane. So, let us add 6 on both sides of the equation to cancel out the -6 and thus, get one numerical value.
- ⇒ y - 6 + 6 = 6x - 8 + 6
- ⇒ y = 6x - 8 + 6
- ⇒ y = 6x - 2
Since 2 is being subtracted from 6x, the y-intercept is (0, -2).